CIVIL 


UNIVERSITY  OF  CALIFORNIA 

DEPARTMENT  OF  CIVIL.  ENGINEERING 

BERKELEY.  CALIFORNIA 


CORRIGENDA 

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The  Theory  of 

GENERAL    RELATIVITY 

and 

GRAVITATION 


Based  on  a  course  of  lectures  delivered  at  the  Conference  on 

Recent  Advances  in  Physics  held  at  the  University 

of  Toronto,  in  January,  1921 


BY 


LUDWIK  SILBERSTEIN,  Ph.D. 
»\ 


NEW  YORK 

D.  VAN  NOSTRAND  COMPANY 

EIGHT  WARREN  STREET 

1922 


. 
•'-- 


Library 


COPYRIGHT,  CANADA.  1922 

BY  THE 
UNIVERSITY  OF  TORONTO  PRESS 


PREFACE 

At  the  Conference  on  Recent  Advances  in  Physics  held 
in  the  Physics  Laboratory  of  the  University  of  Toronto  from 
January  5  to  26,  1921,  a  course  on  Einstein's  Relativity  and 
Gravitation  Theory,  consisting  of  fifteen  lectures  and  two 
colloquia,  was  delivered  by  the  author.  The  first  six  of  these 
lectures  were  devoted  to  what  is  known  as  Special  Relativity, 
and  the  remaining  ones  to  Einstein's  General  Relativity  and 
Gravitation  Theory  and  to  relativistic  Electromagnetism. 
In  view  of  the  time  limitations  only  the  essentials  of  these 
theories  were  dealt  with,  due  attention,  however,  being  given 
to  the  critically  conceptual  side  of  the  subject.  The  Univer- 
sity was  kind  enough  to  undertake  the  publication  of  that 
part  of  the  course  which  dealt  with  general  relativistic  ques- 
tions, on  the  express  understanding  that  my  prospective 
readers  should  be  assumed  to  be  already  familiar  with  the 
special  theory  of  relativity.  In  this  connection  it  was  sug- 
gested by  Prof.  McLennan  that  those  unacquainted  with  the 
older  theory  should  be  referred  to  my  book  of  1914  (The 
Theory  of  Relativity,  Macmillan,  London)  and  that  it  would 
therefore  be  desirable  to  make  the  present  volume,  as  much 
as  possible,  uniform  in  exposition  and  style  with  that  work. 
With  such  requirements  in  view  this  little  book  was  shaped, 
only  a  few  pages  at  the  beginning  having  been  used  in  re- 
calling the  essentials  of  the  special  relativity  theory. 

The  treatment,  as  compared  with  the  Toronto  lectures, 
has  been  made  somewhat  more  systematic  and  the  subject 
matter  has,  here  and  there,  been  considerably  extended. 
In  this  respect  the  author  has  been  partly  influenced  by  a 
larger  course  on  Relativity,  Gravitation  and  Electromagnetism 
delivered,  in  the  time  of  writing,  during  the  last  Summer 
Quarter  at  the  University  of  Chicago.  Such  is  especially 
the  case  with  Chapter  III  in  which  care  has  been  taken  to 
give  the  readers  a  systematic  exposition  of  the  calculus  of 
generally  covariant  beings  called  Tensors.  The  exposition 
follows  here  mainly  upon  Einstein's  own  presentation  of  the 
subject,  with  the  difference,  however,  that  due  emphasis 
has  been  laid  upon  the  distinction  between  metrical  and  non- 
metrical  properties  of  tensors.  But  even  in  this  chapter 


494261 


technicalities  have  been  avoided,  stress  being  laid  upon  the 
guiding  principles  of  this  new,  or  rather  newly  revived,  and 
most  powerful  mathematical  method.  It  seems  hard  to  say 
whether  Einstein's  admirable  theory  has  or  has  not  a  long 
life  before  it  in  the  domain  of  Physics  proper.  But  indepen- 
dently of  its  fate  the  time  applied  for  studying  the  Tensor 
Calculus  and  acquiring  some  skill  in  handling  it  will  be  well 
spent. 

The  plan  of  the  remainder  of  the  book  will  be  sufficiently 
clear  from  the  titles  of  the  chapters  and  sections  arrayed  in 
the  table  of  Contents.  Such  matter  as  seemed  for  the  present 
too  speculative  and  controversial  has  been  relegated  to  the 
Appendix  where,  however,  also  some  points  concerning  the 
curvature  properties  of  a  manifold  have  found  their  place, 
not  only  as  a  preparatory  to  Einstein's  cosmological  specu- 
lations but  perhaps  as  a  useful  supplement  to  Chapter  III. 

The  book  is  felt  to  be  far  from  being  complete.  But  as 
it  is,  it  is  hoped  that  it  will  give  the  reader  a  good  insight  into 
the  guiding  spirit  of  Einstein's  general  relativity  and  gravita- 
tion theory  and  enable  him  to  follow  without  serious  diffi- 
culties the  deeper  investigations  and  the  more  special  and 
extended  developments  given  in  the  large  and  rapidly  growing 
number  of  papers  on  the  subject. 

Some  of  my  readers  will  miss,  perhaps,  in  this  volume 
the  enthusiastic  tone  which  usually  permeates  the  books  and 
pamphlets  that  have  been  written  on  the  subject  (with  a 
notable  exception  of  Einstein's  own  writings).  Yet  the 
author  is  the  last  man  to  be  blind  to  the  admirable  boldness 
and  the  severe  architectonic  beauty  of  Einstein's  theory. 
But  it  has  seemed  that  beauties  of  such  a  kind  are  rather 
enhanced  than  obscured  by  the  adoption  of  a  sober  tone 
and  an  apparently  cold  form  of  presentation. 

My  thanks  are  due  to  Sir  Robert  Falconer  and  to  Prof. 
J.  C.  McLennan  for  promoting  the  cause  of  this  publication, 
to  Prof.  R.  A.  Millikan  and  Prof.  Henry  G.  Gale  of  the 
University  of  Chicago  for  reading  part  of  the  proofs,  and  to 
the  University  of  Toronto  Press  for  the  care  bestowed  on 
my  work. 

L.  S. 
Rochester,  N.Y. 

November  1921. 


CONTENTS 

CHAPTER  I 

Special  Relativity  recalled.    Foundations  of  General  Relativity  and 
Gravitation  Theory 

PAGE 

1.  Inertial  reference  systems ,  1 

2.  Special  relativity  principle 2 

3.  Principle  of  constant  light  velocity 2 

4.  Lorentz  transformation.    Galilean  line-element.    Minimal  lines 

and  geodesies  representing  light  propagation  and  motion  of 

free  particles 4 

5.  Transition  to  general  relativity  and  gravitation  theory.    Infini- 

tesimal equivalence  hypothesis  and  local  coordinates <  9 

6.  Gaussian  coordinates,  and  the  general  line-element 14 

7.  Light  propagation  and  free-particle  motion  expressed  by  the 

general  null-lines  and  geodesies 17 

CHAPTER  II 

The  General  Relativity  Principle.    Minimal  Lines  and  Geodesies. 
Examples.    Newton's  Equations  of  Motion  as  an  Approximation 

8.  Principles  of  general  relativity;  general  covariance  of  laws.  ...         22 

9.  Local  and  system- velocity  of  light 25 

10.  Developed  form  of  geodesies.     Christoffel  symbols 26 

lOa.  First  example:  galilean  system 29 

lOb.  Second  example :  rotating  system 30 

11.  Geodesies,  and  Newtonian  equations  of  motion  as  an  approxi- 

mation           35 

CHAPTER  III 
Elements  of  Tensor  Algebra  and  Analysis 

12.  Introductory.     Gaussian  coordinates 39 

13.  Contravariant  and  covariant  tensors  of  rank  one  or  vectors ...          40 

14.  Inner  or  scalar  product  of  two  vectors.     Zero  rank  tensors, 

invariants 42 

15.  Outer  product.      Tensors  of  rank  two,  symmetrical  and  anti- 

symmetrical.     Mixed  tensors 43 

16.  Tensors  of  any  rank v.         46 

17.  Contraction.     Intrinsic  invariants 46 

18.  Inner  multiplication.     Differentiation  of  tensors 48 

19.  Tensor  properties  in  a  metrical  field.    Quadratic  differential  form 

or  line-element .  .  .^-.;.r^  .^ 50 

20.  Fundamental  tensor.     Metrical  properties  of  tensors.     Norm 

and  size.     Conjugate  tensors 52 

21.  Supplement.     Reduced  tensor 55 

22.  Angle  and  volume.     Sub-domains 56 


PAGE 

23.*Differentiation  based  on  metrics.  Covariant  derivative  or  ex- 
pansion; contra  variant  derivative.  Rotation  of  a  vector. 
Antisymmetric  expansion  of  a  six-vector.  Divergence  of  a 
six-and  of  a  four-vector 59 

24.  The  Riemann-Christoffel  tensor.     Riemannian  symbols  and 

curvature.     Lipschitz's  theorem 62 

CHAPTER  IV 
The  Gravitational  Field-equations,  and  the  Tensor  of  Matter 

25.  Contracted  curvature  tensor.     Einstein's  field-equations  outside 

^        of  matter.     Bianchi's  identities 69 

26.  Laplace's  equation,  and  Newton's  law,  as  a  first  approximation        72 

27.  The  tensor  of  matter.    Einstein's  field-equations  within  matter. 

Laplace-Poisson's  equation  as  a  first  approximation.    Mean 
curvature  and  density  of  matter.     Example 74 

28.  Equations  of  Matter.     The  principles  of  momentum  and  of 

energy.     Remarks  on  conservation 81 

29.  Hamiltonian  Principle 88 

30.  Gravitational  waves.    Einstein's  approximate  integration  of  the 

field-equations 90 

CHAPTER  V 

Radially  Symmetric  Field.    Perihelion  Motion,  Bending  of  Rays, 
and  Spectrum  Shift 

31.  Radially  symmetrical  solution  of  the  field-equations 92 

32.  Perihelion  of  a  planet.     Mercury's  excess 95 

33.  Deflection  of  light  rays.    Results  of  the  Sobral  Eclipse  Expedi- 

tion        100 

34.  Shift  of  spectrum  lines.    The  atoms  as  'natural  clocks' 102 

CHAPTER  VI 

35.  Generally  covariant  form  of  the  equations  of  the  electromag- 

netic field 106 

36.  The  four-potential Ill 

37.  Orthogonal  curvilinear  coordinates 113 

38.  Propagation  of  electromagnetic  waves  in  a  gravitational  field ..  115 

39.  Ponderomotive  force  and  energy  tensor  of  the  electromagnetic 

field 119 

APPENDIX 

A.  Manifolds  of  Constant  Curvature 124 

B.  Einstein's  New  Field-Equations  and  Elliptic  Space 129 

C.  Space-Time  according  to  de  Sitter 135 

D.  Gravitational  Fields  and  Electrons 137 

Index .  .  138 


CHAPTER  I. 

jcial  Relativity  recalled.     Foundations  of  General 
Relativity  and  Gravitation  Theory. 


In  accordance  with  the  purpose  and  the  origin  of  this 
volume*  its  readers  are  assumed  to  have  already  made  them- 
selves familiar  with  the  essentials  of  Einstein's  older  or  special 
Relativity.  It  will  be  enough,  therefore,  to  recall  here  very 
concisely  what  of  that  theory  may  be  conducive  to,  and  even 
necessary  for,  a  thorough  grasping  of  the  structure  and  the 
aims  of  the  more  general  theory,  and  of  the  spirit  pervading  it. 

1.  First  of  all,  then,  out  of  all  thinkable  reference-frame- 
works, the  special  relativity  is  concerned  only  with  a  certain 
privileged  class  of  frameworks  or  systems  of  reference,  the 
inertiol  systems.  Of  these  there  are  <»3.  If  5,  say  the 
'fixed-stars'  system,  is  one  of  them,  any  other  rigid  system  Sr 
of  coordinate  axes  moving  relatively  to  S  with  any  uniform 
and  purely  translational  velocity  v,  in  any  direction  whatever, 
is  again  an  inertial  system  or  belongs  to  the  same  privileged 
class.  And  the  systems  thus  derived  from  5,  or  from  one 
another,**  exhaust  the  class.  Since  the  size  or  absolute 
value  of  the  relative  velocity  implies  one  scalar  datum,  and  its 
direction  two  more  such  data,  all  independent  of  one  another, 
there  is  just  a  triple  infinity  of  inertial  systems,!  as  already 
stated.  Not  that  the  special  relativity  theory  abstains  from 
considering  accelerated,  i.e.  non-uniform  motion  of  particles 
within  any  of  these  systems;  but  it  does  not  contemplate  any 
frameworks  other  than  the  inertial  ones  as  systems  of  refer- 

*Cf.  Preface. 

**If  Sf  moves  uniformly  with  respect  to  5,  and  5"  w;th  respect  to  Sf,  so 
does  S"  with  respect  to  S.  If  the  reader  so  desires,  he  may  consider  this  as 
a  postulate. 

'  fThe  purely  spatial  orientation  of  the  axes,  implying  further  free  data, 
is  irrelevant  in  the  present  connection. 

1 


*  «T     3  »     «   O^  •        *-J          a*     f  % 

RELATIVITY  AND  GRAVITATION 

ence,  and  cannot,  nor  does  it  propose  to  deal  with  them.  It 
is  unable,  for  instance,  to  transform  the  course  of  phenomena 
from  the  S  system  to  the  spinning  Earth  or  to  an  accelerated 
carriage  as  reference  systems. 

2.  Keeping  this  in  mind,  the  first  main  assumption  of  the 
older  theory,  known  as  the  Special  Relativity  Principle,  can 
be  briefly  stated  by  saying  that  it  requires  the  laws  of  physical 
phenomena  to  be  the  same  whether  they  are  referred  to  one  or 
to  any  other  inertial  system.     In  short,  the  maxim  of  the 
1905 — Relativity  was:     Equal  laws  for  all  inertial  systems. 

The  italicized  words  are,  mathematically  speaking,  at 
first  somewhat  vague.  In  fact,  they  are  intended  to  stand 
for  'the  same  form  of  mathematical  equations  expressing  the 
laws.'  Now,  since  this  implies  the  use  of  some  magnitudes, 
such  as  the  coordinates  and  the  time,  or  the  electric  and  the 
magnetic  vectors  (forces),  in  each  of  the  said  systems,  the 
requirement  of  mathematical  'sameness'  remains  cloudy  until 
we  are  told  what  dictionary  is  to  be  used  to  translate  the 
language  of  one  into  that  of  any  other  inertial  system,  or 
technically,  to  transform  from  the  non-dashed  to  the  dashed 
variables.  This  vagueness,  however,  soon  disappears,  giving 
place  to  precision,  in  the  next  fundamental  step  of  the  theory 
as  will  be  seen  presently. 

The  attentive  reader  might  here  object  by  saying  that  'sameness  of 
laws'  means  absence  of  difference,  absence  of  observable  different  behaviour 
(of  moving  bodies  or  of  electric  waves)  in  passing  from  an  5  to  an  S',  and 
that,  therefore,  to  begin  with,  no  mathematical  magnitudes  or  equations 
are  required.  But  actually  we  are,  perhaps  forever,  confined  to  one 
(approximately)  inertial  system,  our  planet,  and  are  thus  unable  to 
observe  directly  the  permanence  of  behaviour  in  passing  tp  another 
system  of  reference.  The  only  way  open  to  us  is  to  proceed,  through 
more  or  less  long  chains  of  abstract  reasoning,  from  the  principle  of 
relativity  to  some  observable  prediction,  and  such  processes  are  scarcely 
practicable  without  the  use  of  mathematical  symbols  and  equations. 

3.  The  second  assumption,  called  the  Principle  of  Constant 
Light- velocity,  apart  from  its  own  importance,  provides  for  the 
need  just  explained,  its  true  office  in  the  structure  of  the 
theory  being  to  set  an  example  of  a  'physical  law'  which  is 
postulated    to   satisfy    rigorously    the   first   assumption.     It 


CONSTANT  LIGHT  VELOCITY  3 

runs  thus:  Light  is  propagated,  in  vacua,  relatively  to  any 
inertial  system,  with  a  velocity  c,  constant  and  equal  for 
all  directions,  no  matter  whether  the  source  emitting  it  is 
fixed  or  moving  with  respect  to  that  system.  This  is  shortly 
referred  to  as  uniform  and  isotropic  light  propagation  in  any 
inertial  system.  The  light  velocity,  in  empty  space,  plays  the 
part  of  a  universal  constant, — which  role,  however,  it  will 
readily  give  up  in  generalized  relativity. 

The  reader  is  well  acquainted  with  the  mathematical 
expression  of  the  consequence  of  these  two  assumptions 
(together  with  a  tacit  requirement  of  formal  equivalence  of 
any  two  inertial  systems  S,  Sf),  to  wit,  the  invariance  of  the 
quadratic  form 

c*P-x*-y*-&,  (a) 

where  x,  y,  z  are  the  cartesian  co-ordinates  and  /  the  time  of 
the  5-system.  That  is  to  say,  if  x',  y',  z1 ',  tf  be  the  cartesian 
co-ordinates  and  the  time  used  in  any  other  inertial  system 
5',  (a)  should  transform  into 

cW-x't-yV-z'*.  (a') 

As  a  matter  of  fact,  what  was  originally  required  was  that  the 
equation  (a)  —  0  should  transform  into  (a')  =  0,  and  this 
would  be  satisfied  by  putting  (a')=\  .  (a),  where  X  is  inde- 
pendent of  x,  y,  z,  t  but  might  be  some  function  of  v,  the  relative 
velocity  of  5",  Sf.  This,  however,  would  amount  to  a  dis- 
tinction between  the  two  systems,  at  least  a  formal  one, 
unless  X=l.  If,  therefore,  equal  rights  are  claimed  not 
only  physically  but  also  formally,  mathematically,  for  all 
inertial  systems,  we  have  (a)  =  (a'),  that  is  to  say,  the  quadratic 
form  (a)  is  raised  to  the  dignity  of  an  invariant. 

There  is,  certainly,  nothing  to  object  to  in  such  a  procedure, 
especially  as  it  carries  simplicity  with  itself.  Yet  these 
remarks  did  not  seem  superfluous,  especially  as  there  is  among 
the  relativists  a  strong  tendency  to  a  certain  kind  of  hypostasy 
of  the  said  quadatic  form*  (by  declaring  it  to  be  more 

Intensified  more  recently  in  the  case  of  the  more  general  (differ- 
ential) quadratic  form  playing  a  fundamental  r61e  in  the  newer  relativity 
theory,  as  will  be  seen  hereafter. 


4  RELATIVITY  AND  GRAVITATION 

'objective,  real  or  intrinsic'  than  space-distance  or  time)  just 
because  it  "is"  invariant,  —  and  forgetting  that  we  have 
deliberately  made  it  invariant. 

4.  MawMPtfciie,  returning  to  the  quadratic  expression  (a), 
let  us  write  it  down  for  a  pair  of  events  infinitesimally  near 
to  one  another  in  space  and  time.  Thus,  writing  #lf  x%,  ac3,  x4 
for  x,  y,  z,  ct,  the  statements  made  above  can  be  expressed 
by  saying  that  the  quadratic  differential  form 

ds*  =  dxf-dxi*-dx>i*-dxt  (1) 

should  be  invariant  with  respect  to  the  passage  from  one 
inertial  system  5  to  any  other  such  system  S'.  The  differ- 
ential foim  is  here  preferable  to  the  original  one,  as  it  will  be 
helpful  in  paving  the  way  for  general  relativity. 

As  is  well  known,  this  requirement  of  in  variance  gives  the 
rule  of  transformation  of  the  variables  xt  into  those  x\  of  the 
S'-system,  called  the  Lorentz  transformation.  If  both  the  Xi 
and  the  x'\  axes  are  taken  along  the  line  of  motion  of  5' 
relatively  to  5,  with  the  velocity  v  =  fic,  if  further  the  xz,  xs  — 
axes  are  taken  parallel  to  those  of  #'2,  #'3,  and  if  the  convention 
x'i  =  xfi  =  Q  for  XI  =  XI  =  Q  is  adopted,  the  Lorentz  transforma- 
tion assumes  the  familiar  form 

*'i  =  7(*i—  0*4),  xf2  =  x2,  x'z  =  X3,  x'4  =  y(xi-pxi)        (2) 
where  y  =  (1  —  £2)  ~^.     Vice  versa,  we  have,  by  solving  (2), 


showing  the  complete  (including  the  formal)  equivalence  of  the 
two  systems.  Let  us  keep  well  in  mind,  for  what  is  to  follow, 
that  this  transformation  is  a  linear  one,  with  constant  co- 
efficients, and  that  special  relativity,  concerned  with  inertial 
systems  only,  does  not  contemplate  any  other  space-time 
transformations. 

Every  tetrad  of  magnitudes  XL  (i  =  l  to  4)  which  are 
transformed  as  the  xt,  is  called  a  four-vector  or,  after  Min- 
kowski,  a  world-vector  of  the  first  kind.  Such  four-vectors 
are,  in  addition  to  dxt  or  xt  itself,  their  prototype,  the  four- 
velocity  dxjds  and  the  four-acceleration  of  a  moving  particle, 
the  electric  four-current,  and  so  on.  To  every  vector  Xl 


LORENTZ  TRANSFORMTAION  5 

belongs  a  scalar  or  invariant  Xf—X^—X'f^-X^,  its  only 
invariant  with  respect  to  the  Lorentz  transformation.  But 
we  need  not  stop  here  to  reconsider  the  properties  of  the  four- 
vector  and  other  world-vectors,  such  as  the  six-  vector,  which 
constituted  the  only  lawful  material  of  the  older  relativity 
for  writing  down  laws  of  Nature,  —  especially  as  we  shall  soon 
return  to  these  mathematical  entities  as  particular  cases  of 
tensors  of  various  ranks  which  are  indispensable  to  the  general 
theory  of  relativity. 

On  the  other  hand  we  may  profitably  dwell  yet  a  while 
upon  the  quadratic  form  (1)  itself,  the  square  of  the  line- 
element,  as  ds  is  called.  Granted  the  assumptions  of  special 
relativity,  this  expression  becomes  the  fundamental  quadratic 
differential  form  of  the  four-manifold,  the  world  or  space  - 
time,  in  exactly  the  same  way  as 


is  the  fundamental  form  of  a  flat  two-space  or  surface,  and 
more  generally, 

da2  =  Edu*+  2  Fdudv  +  Gdv2 

that  of  any  surface,  and 

da2  =  dr2+R2  sin2  ^  (sin2  0  d82+d<j>2) 

the  fundamental  differential  form  of  any  three-space  of  con- 
stant curvature  R~2.~\  Now,  it  is  enough  to  open  any  book 
on  differential  geometry  to  see  that,  with  the  usual  assump- 
tions of  continuity,  etc.,  the  whole  geometry,  i.e.,  all  metrical 
properties  of  the  two-space  or  the  three-space  in  question 
are  completely  determined  by  the  corresponding  differential 
forms.  Their  geodesies  or,  within  restricted  regions  at  least, 
their  shortest  lines,  the  angle  relations,  and  their  whole 
trigometry,  all  this  is  fully  determined  provided  the  co- 
efficients of  the  differentials,  such  as  E,  etc.,  appearing  in 


fAccording  as  R*  is  positive,  zero  or  negative,  we  have  an  elliptic, 
lidean  (or  parabolic)  or  hyperbolic  s 
becomes  R  sinh  (r/R),  where  R?  =  —  R?. 


euclidean  (or  parabolic)  or  hyperbolic  space.      In  the  latter  case  R  sin    - 

JK. 


6  RELATIVITY  AND  GRAVITATION 

the  fundamental  form  are  given  functions  of  the  variables.* 
This  deterministic  mastery  of  the  quadratic  differential 
form  has  been,  as  far  back  as  1860,  technically  extended  to 
spaces  or  manifolds  of  four  and,  in  fact,  of  any  number  of 
dimensions, — although,  not  being  sufficiently  sensational,  it 
never  attracted  the  attention  of  anybody  beyond  a  few 
specialists. 

In  much  the  same  way  all  the  metrical  properties  of  the 
four-dimensional  world  of  the  special  relativist  sjiould  be, 
and  are,  derivable  from  the  fundamental  form  (1)  belonging, 
or  rather  allotted  to  it.  This  is,  from  the  point  of  view  of  the 
poly-dimensional  differential  geometer,  but  a  very  special,  in 
fact,  the  most  simple  quadratic  form  in  four  variables.  For 
it  contains  but  the  squares  of  their  differentials,  and  the 
coefficients  of  these  are  all  constant,  which — in  view  of  the 
sequel — it  may  be  well  to  bring;  into  evidence  by  writing  (1) 

ds2  =  g^dx.dx.,  (la) 

to   be    summed    over    i,    ic  =  l,    2,    3,    4,    tabulating   the   co- 
efficients, thus 

-1000 

0-100  (Ib) 

0  0-1  0 

0001 

and  calling  this  array  of  special  coefficients  the  inertial  or 
the  galilean  glK.  We  shall  denote  them  in  the  sequel  by  gt(C . 
To  give  this  array  is  as  much  as  to  give  the  form  (la),  and 
herewith  the  properties  of  the  world, — for  it  is  manifestly 
irrelevant  how  we  call  or  denote  the  four  corresponding 
variables.  The  values  of  the  gtK  being  given,  the  properties 

*To  be  rigorous  we  should  have  said  '  all  properties  of  a  restricted  region 
of  the  contemplated  manifold';  for  certain  properties  of  the  manifold  as 
a  whole  are  still  left  free.  The  choice,  however,  is  limited  to  a  small  number 
of  discrete  possibilities.  Thus,  for  example,  there  are  two  kinds  of  elliptic 
space,  the  spherical  or  antipodal,  and  the  polar  or  elliptic  proper.  In  the 
former  the  total  length  of  a  straight  line  (geodesic)  is  2irR,  and  in  the  latter 
TrR;  the  planes  are  two-sided,  and  one-sided,  respectively,  and  so  on. 


MINIMAL  LINES  AND  GEODESICS  7 

of  the  x  will  follow  by  themselves.  There  is  no  need  to 
declare  beforehand  that  they  are  cartesian  coordinates  of  a 
place  and  its  date.  Further,  the  circumstance  that  these 
coefficients  are  of  different  signs,  three  being  negative,  and 
one  positive,  creates  for  the  general  geometer  no  difficulty. 

This  circumstance  brings  only  with  it  the  important 
feature  that  there  are  in  the  world  real  minimal  lines*  as  the 
geometer  would  put  it,  that  is  to  say,  lines  of  zero-length, 


or 


These  special  world  lines  represent  the  propagation  of  light 
or,  apart  from  physical  difficulties,  the  uniform  motion  of  a 
particle  with  light  velocity  c.  As  a  matter  of  fact  the  very 
first  step  of  the  theory  consisted  in  writing  ds  =  Q  as  the  ex- 
pression of  light  propagation  in  vacuo. 

In  the  next  place  consider  the  equally  fundamental  con- 
cept of  the  geodesies  of  the  world.    These  are  defined  by 


the  limits  of  the  integral  being  kept  fixed.  To  derive  from 
this  variational  equation  the  differential  equations  of  a 
geodesic,  proceed  in  the  well-known  way.  If  u  be  any  inde- 
pendent parameter,  and  if  dots  are  used  for  the  derivatives 
with  respect  to  it,  we  have 

8fsdu  =/&>  .du  =  Q, 

where,  by  (1),  s2  =  —  (xf+x£-\-xJ-)-\-x?,  and  therefore, 
ds  =   -    XtdXi 


*Whereas  on  any  (real)  surface  all  the  4  minimal  lines '  (known  also  as 
null-lines),  which  play  in  the  surface  theory  an  important  analytical  r61e, 
are  always  imaginary.  The  reader  will  do  well  to  consult  on  this  and  allied 
topics  a  special  treatise  on  differential  geometry. 


8  RELATIVITY  AND  GRAVITATION 

By  partial  integration,  and  remembering  that  all  8xt  vanish 
at  the  limits  of  the  integral, 

I  -  xtdxt  .du=-  \—(-  x\xt  .  du. 
j  s  J  du\  s     / 

Thus,    the   dxt  being   mutually   independent,    the   required 
differential  equations  are 


If  the  geodesic  does  not  happen  to  be  a  null-line  (light  pro- 
pagation) we  can  as  well  take  u  =  s,  when  $=1,  and  the 
equations  become 


whence 


dxt       dt 

—  = =  a  =  const. 

ds       dx±  ds 


The  fourth  of  these  equations  is  dx*/ds  =  const. ,  and  therefore, 
the  first  three, 

dt  dt          '   dt 

and  these  represent  uniform  rectilinear  motion,  which  is  the 
motion  of  a  free  particle. 

Let  us,  therefore,  keep  well  in  mind  these  two  properties 
of  the  line-element  ds  of  special  relativity: 

I.  The  minimal  lines  of  the  world, 

ds  =  0,  (I) 

represent  light  propagation  in  vacua. 

II.  The  world  geodesies,  defined  by 

d/ds  =  Q,  (II) 

with  fixed  integral  limts,  represent  the  motion  of  a  free  particle. 


MINIMAL  LINES  AND  GEODESICS  9 

A  special  emphasis  is  here  put  on  these  two  properties 
because  they  will  be  carried  over  to  the  general  relativity 
and  gravitation  theory,  and  because  these  and  principally 
only  these  two  properties  constitute  the  connection  of  the 
otherwise  purely  analytical  differential  form  ds2  =  gucdxldxK 
with  physics.  In  other  words,  (I)  as  the  equation  of  light 
propagation,  and  (II)  as  that  of  the  motion  of  a  free  particle 
impart  physical  meaning  to  the  mathematical  form  which 
is  the  'line-element'  ds.  Without  this  all  the  properties  of 
the  quadratic  form,  though  interesting,  perhaps,  in  them- 
selves, would  have  nothing  to  do  with  the  world  of  physical 
phenomena. 

It  is  scarcely  necessary  to  say  that  the  law  (II)  of  the 
motion  of  free  particles  is,  as  well  as  (I)  for  light,  invariant 
(thus  far)  with  respect  to  the  Lorentz  transformation.  For 
it  is,  by  its  very  structure,  independent  of  the  choice  of  a 
reference  system  S.  Since  ds  is  invariant,  so  isfds,  extended 
between  any  two  world-points.  Thus  also  the  developed 
form  of  (II),  the  system  of  differential  equations  dzxJds*  =  Q, 
is  transformed  in  Sf  into  d*x,f/ds2  =  Q.  And  in  fact,  uniform 
motion  of  a  particle  relatively  to  5,  means  also  (originally  by 
an  assumption)  its  uniform  motion  in  any  other  inertial 
system  S'.  In  short,  the  Lorentz  transformation  leaves  the 
uniformity  of  motion  of  a  particle  intact. 

5.  We  are  now  ready  to  pass  to  Einstein's  theory  of 
general  relativity  and  gravitation.  Not  that  our  task  is  an 
easy  one,  but  we  are  somewhat  better  prepared  to  embark 
upon  it. 

Why  equal  form  of  physical  laws,  why  equal  rights  for 
the  inertial  systems  only?  Why  not  equal  rights  for  all 
(systems)?  Such  would  be  the  urgent,  and  yet  vague,  ques- 
tions naturally  suggesting  themselves  after  what  was  said 
in  the  preceding  sections.  Yet  it  is  not  with  these  questions, 
nor  with  an  attempt  to  answer  them,  that  we  will  begin  our 
journey  across  this  new  and  revolutionary  country.  For, 
even  if  answered,  these  questions  would  remain  physically 
barren  were  it  not  for  the  existence  of  gravitation  and 


10  RELATIVITY  AND  GRAVITATION 

especially  of  a  certain  peculiarly  simple  property  of   this 
universal  agent. 

This,  therefore,  will  first  occupy  our  attention  for  a  while. 
The  cardinal  feature  of  gravitation  just  hinted  at  is  the  pro- 
portionality of  weight  to  mass,  in  other  words,  the  proportion- 
ality of  heavy  (gravitating)  and  inert  mass.  First  tested  by 
Newton  in  his  famous  pendulum  experiments  with  bobs  of 
different  material,  and  carried  to  further  precision  by  Bessel, 
this  proportionality  has  been  more  recently  shown  by  Roland 
Eotvos  to  hold  to  one  part  in  ten  millions.  It  is  reasonable, 
therefore,  to  assume,  with  Einstein,  that  it  holds  rigorously,* 
at  least  until  proofs  to  the  contrary  are  forthcoming.  In  our 
present  connection  it  is  better  to  express  this  property  more 
directly  by  saying,  even  with  Galileo,  that  all  bodies,  light 
or  heavy,  fall  equally  in  vacuo.  All  particles,  that  is,  acquire 
at  a  given  place  of  a  gravitational  field  equal  accelerations 
independently  of  their  own  mass  or  chemical  nature,  etc.,  and 
no  matter  how  much  of  their  inertia  is  due  to  the  energy 
stored  in  them  and  how  much  of  other  origin.  This  remark- 
able property  distinguishes  the  gravitational  field  from  other 
fields.  Take,  for  instance,  an  electric  field  given  by  the  vector 
E.  The  force  on  a  particle  of  rest-mass  m,  carrying  the 
electric  charge  e,  and  starting  from  rest,  is  eE,  and  the  accelera- 
tion eE/m.  Now,  in  general,  there  is  no  relation  between  m 
and  e,  and  even  if  the  mass  is  purely  electromagnetic,  when  m 
is  proportional  to  ez/a,  the  acceleration  will  vary  from  particle 
to  particle  inversely  as  its  charge  and  directly  as  its  average 
diameter,  2a.  We  have  disregarded,  of  course,  the  dielectric 
properties  of  the  particle  which  would  make  its  behaviour 
in  a  given  electric  field  still  more  complicated.  The  same 
remarks  would  hold,  mutatis  mutandis,  for  the  behaviour  of 
different  bodies  placed  in  a  magnetic  field.  In  short,  gravita- 
tion is,  in  this  respect,  unique  in  its  simplicity. 


*In  a  theory  of  matter  and  gravitation  proposed  by  G.  Mie,  Annalen 
der  Physik,  vols.  37,  39,  40  (1912  and  1913),  the  proportionality  between 
weight  and  mass  does  not  hold  rigorously,  though  to  an  order  of  precision 
much  exceeding  that  stated  by  Eotvos. 


EQUIVALENCE  HYPOTHESIS  11 

This  very  circumstance  enabled  Einstein  to  undertake  his 
mental  experiment  with  the  falling  or  ascending  elevator, 
now  so  familiar  to  the  general  public.  In  fact,  consider  a 
homogeneous  or  a  quasi-homogeneous  gravitational  field 
such  as  the  terrestrial  one  in  a  properly  restricted  region.  Let 
a  lift  or  elevator,  small  compared  with  the  earth,  yet  ample 
enough  for  a  physical  laboratory  and  for  those  in  charge  of 
it,  descend  vertically  with  the  local  terrestrial  acceleration  g. 
Then  all  bodies  placed  anywhere  within  the  elevator  and 
left  to  themselves  will  float,  in  mid-air  or  better  in  vacuo, 
and  particles  projected  in  any  direction  will  move  uniformly 
in  straight  paths  relatively  to  the  elevator.  Moreover,  all 
objects,  including  the  physicists,  standing  or  lying  about 
will  cease  to  press  against  the  floor  or  the  tables,  as  the  case 
may  be.  In  short,  all  traces  of  gravitation  will  be  gone,* 
and  the  inmates  of  the  lift,  assumed  to  have  no  intercourse 
whatever  with  the  outer  world,  will  declare  their  reference 
system  to  be  a  genuine  inertial  system, — so  far,  at  least,  as 
mechanical  phenomena  are  concerned.  For  an  unbiassed 
judge  cou  Id  not  tell  beforehand  whether  it  will  be  also  optically 
inertial,  that  is  to  say,  whether  the  law  of  constant  light 
velocity  will  hold  good  for  the  lift.  Einstein  thinks  it  will,  or 
rather  assumes  it,  more  or  less  implicitly.  If  this  be  granted, 
we  can  say  that  the  elevator  will  be  an  inertial  reference 
system  in  every  respect. 

The  possibility  of  thus  undoing  a  gravitational  field  is 
manifestly  based  on  the  said  equal  behaviour  of  all  bodies 
placed  in  it.  For  otherwise  the  artificial  motion  of  the  elevator 
could  not  be  adapted  to  all  bodies  at  the  same  time,  each  of 
them  requiring  a  different  acceleration. 

Next,  pass  to  any,  non-homogeneous  gravitation  field, 
which  in  the  most  general  case  may  also  vary  with  time.  This 
certainly  cannot  be  undone,  as  a  whole,  by  a  single  elevator 
as  reference  system.  But  you  can  imagine  an  ever  increasing 
number  of  sufficiently  small  elevators,  each  appropriately 
accelerated,  fitted  into  small  regions  of  the  field,  and  each, 

*Vice  versa,  in  absence  of  a  gravitational  field,  a  lift  in  accelerated 
ascending  motion  would  give  us  a  faithful  imitation  of  such  a  field. 

—2 


12  RELATIVITY  AND  GRAVITATION 

perhaps,  to  do  its  duty  for  a  very  short  interval  of  time,  and 
to  be  replaced  by  another  in  the  next  moment.  These  minute 
elevators  will  do  their  office  at  least  in  the  mechanical  sense 
of  the  word.  Einstein  assumes  that  they  will  act  as  inertial 
systems  also  in  the  optical  sense  of  the  word,  as  explained 
above.  This  process  of  subdividing  a  gravitational  field,  in 
space  and  time,  and  fitting  in  of  appropriate  small  elevators 
can  be  carried  on  to  any  required  degree  of  approximation. 

In  fine,  passing  to  the  limit,  let  us  make,  with  Einstein,* 
the  explicit  assumption: 

With  an  appropriate  choice  of  a  local  reference  system 
(«i,  u2,  Us,  u^)  special  relativity  holds  for  every  infinitesimal 
four -dimensional  domain  or  volume- element  of  the  world. 

That  is  to  say,  at  every  world-pointf  a  system  of  space- 
time  coordinates  u\,  u%,  u3,  u±  can  be  chosen  in  which  the  line- 
element  assumes  the  galilean  form 

ds2  =  du?  -  dui*  -  du^  -  du^.  (3) 

These  four  coordinates  are  called  local  coordinates.  With 
respect  to  this  local  system  there  is  then  no  gravitational  field 
at  the  given  world-point,  and  in  accordance  with  special 
relativity  ds2  has^ there  a  value  independent  of  the  'orienta- 
tion '  of  the  local  axes;  that  is  to  say,  the  quadratic  form  (3) 
is  invariant  with  respect  to  the  Lorentz  transformation  (2) . 

It  is  this  assumption  which  can  now  be  properly  referred 
to  as  the  infinitesimal  equivalence  hypothesis,  for  it  grew  out 
of  Einstein's  original  equivalence  hypothesis  applied  to  finite 
regions  when,  in  his  first  attempt  at  a  theory  of  gravitation 
(1911),  he  was  confining  himself  to  a  homogeneous  field. 

Whatever  the  origin  of  this  hypothesis  or  assumption,  it  is 
certainly  not  difficult  to  adhere  to  it.  For  it  scarcely  amounts 
to  anything  more  than  to  assuming,  in  the  case  of  a  curved 
surface,  say,  the  existence  of  a  tangential  plane  at  any  of  its 


*A.  Einstein,  Die  Grundlagen  der  allgemeinen  Relativitatstheorie . 
Annalen  der  Physik,  vol.  49,  1916,  p.  777. 

fWith  the  possible  exception  of  some  discrete  points,  such  perhaps 
as  those  at  which  the  density  of  matter  acquires  enormous  values. 


INFINITESIMAL  WORLD  FLATNESS  13 

points,  or  to  declare  the  surface  to  be  (in  Clifford's  termino- 
logy) elementally  flat.  And  it  will,  perhaps,  be  well  to  restate 
shortly  Einstein's  hypothesis  by  saying  that  it  assumes  the 
four-dimensional  world  to  be,  in  presence  as  well  as  in  absence 
of  gravitation,  elementally  flat.  It  will  not  be  forgotten,  how- 
ever, that  this  geometric  term  is  nothing  more  than  a  synonym 
of  elementally  galilean,  i.e.,  satisfying  special  relativity  in- 
finitesimally.* 

To  avoid  the  danger  of  any  misconception  let  us  dwell 
upon  this  subject  yet  for  a  while.  The  coordinates  ut  with 
their  corresponding  galilean  line-element  (3)  were  set  up 
only  for  a  local  purpose,  their  real  office  being  confined  to  a 
fixed  world-point  P,  say  Xi,  Xz,  x3,  #4  (in  any  coordinate 
system).  If  we  so  desire,  we  may  think  of  a  whole  galilean 
world  [/determined  throughout,  to  any  extent,  by  the  simple 
form  (3) .  But  as  a  tangential  plane  has  something  in  common 
with  a  surface  only  at  the  point  of  contact  and  then  diverges 
from  it,  ceasing  to  represent  any  intrinsic  properties  of  the 
surface  itself,  so  has  the  auxiliary  and  fictitious  world  U 
anything  to  do  with  the  actual  world  W  (complicated  by 
gravitation)  at  the  world  point  xt  only.  The  fictitious  world 
U  is  tangential  to  the  actual  world  W  at  that  point,  and  parts 
company  with  it  beyond  the  point  of  contact.  At  other 
world-points  the  role  of  U  is  taken  over  by  other  and  other 
fictitious  galilean  worlds.  One  more  cautious  remark.  The 
contact  of  U  and  W  is  one  of  the  first  order,  i.e.,  such  as 
the  contact  between  a  surface  and  its  tangential  plane 
or  between  a  curve. and  its  tangential  line,  but  not  as  the 
more  intimate  contact  bewteen  a  curve  and  its  circle  of 
curvature  (which  is  of  the  second  order).  This  circumstance 
may  acquire  some  importance  later  on. 


*As  to  the  concept  of  elementary  flatness  of  a  surface  or  a  more-dimen- 
sional space,  it  is  beautifully  explained  in  W.  K.  Clifford's  '  Philosophy  of 
Pure  Sciences',  published  in  his  famous  Lectures  and  Essays  (Macmillan, 
London).  Notice  that  in  Clifford's  sense  every  regular  surface,  no  matter 
how  curved,  is  elementally  flat,  with  the  exception  of  some  singular  points, 
such  as  the  vertex  of  a  cone. 


14  RELATIVITY  AND  GRAVITATION 

6.  Having  thus  made  clear  the  local  character  of  the  wt 
coordinates,  let  us  now  introduce  any  coordinate  system  xt 
whatever,  to  be  used  as  a  reference  system  of  coordinates 
for  the  whole  world,  i.e.,  throughout  the  gravitational  field 
and  through  all  times.  Then,  if  xt  be  the  reference  coordinates 
of  P,  and  xt  +  dxt  those  of  a  neighbour-point  ut  +  dut,  the 
differentials  dut  will  in  general  be  linear  homogeneous  func- 
tions of  all  the  dxt,  say 

4 

duL  =  S  au  dxK, 

K=l 

or  with  the  conventional  abbreviation, 

dut  =  du,.  dxK,  (4) 

where  the  coefficients  a^  will  in  general  be  functions  of  all 
the  xt.  It  is  of  importance  to  note  that  that  the  relations  (4) 
will,  generally  speaking,  be  not-integrable,  or  borrowing  a 
name  from  dynamics,  non-holonomous,  that  is  to  say,  the 
a^  will  not  necessarily  be  dut/dxK,  the  differential  expressions 
on  the  right  of  (4)  will  not  be  total  differentials  of  functions 
of  the  xt,  and  there  will  be  no  finite  relations  between  the 
local  and  the  general  or  reference-coordinates. 

Substituting  (4)  into  (3),  collecting  the  terms  and  calling 
gut  the  coefficient  of  the  product  of  differentials  dxtdxK  we 
shall  have,  for  the  line-element  in  the  general  reference 
system, 

ds2  =  glK  dxt  dxK,  (5) 

where  guc  =  glct  will,  in  the  most  general  case,  be  functions  of 
all  the  xt.  But  since  ds*,  as  defined  originally  by  (3),  was  in- 
dependent of  the  orientation  of  the  local  system  of  axes,  so 
also  will  the  ten  different  coefficients  gtK,  though  functions  of 
the  coordinates  xt,  be  manifestly  independent  of  the  orienta- 
tion of  the  local  system. 

The  line-element  will  thus  be  represented,  in  any  reference 
coordinates  xt  whatever  by  the  most  general  quadratic  differen- 
tial form  of  these  four  variables,  such  in  fact  being  the  form 
(5).  As  before  the  summation  sign  is  omitted;  the  sum- 
mation is  to  be  extended  over  t,  K  from  1  to  4,  each  of  these 


GENERAL  TRANSFORMATIONS 


15 


suffixes,    i    and    /c,    appearing    twice.     Thus, 

2gi2dxidx2  +  .  .  .  -\-gudxf.      The  reader  will   soon   learn   to 

handle  this  abbreviated  and  very  convenient  symbolism. 

Suppose  now  we  introduce  instead  of  xt  any  other  set  of 
space-time  coordinates  xt',  any  functions  whatever  of  the  xt, 
such,  that  is,  that  between  the  two  sets  exist  any  given  holo- 
nomous  relations 

#i  =  &  (*i'i  #2',  #3',  xi),  (6) 

the  </>t  being  any  functions  whatever,  but  continuous  together 
with  their  first  derivatives  and  such  that  their  Jacobian,  the 
well-known  determinant 

xi    dxi          dx\ 


J  = 


(7) 


d*4' 
does  not  vanish.    Under  these  circumstances  we  have 


and  vice  versa, 


dxK 


(8) 


(8a) 


and,  as  it  may  be  well  to  notice  in  passing,  //'  =  !,  where  /' 


is  the  inverse  Jacobian 


dx! 

dxK 


Now,  substituting  (8)  into 


(5),  gathering  again  the  terms,  and  denoting  the  coefficient  of 
dxt'dxK'  by  glKr,  i.e.,  putting 

'  =  d^q  dffff  /g\ 

we  shall  have 

which  is  (5)  reproduced  in  dashed  letters.     Not  that  the  &/ 
will  be  functions  of  the  #/  of  the  same  form  as  were  the  gu  of 


16  RELATIVITY  AND  GRAVITATION 

xlt  but  only  that  the  quadratic  differential  form  remains 
quadratic.  There  is  certainly  nothing  surprising  in  this  kind 
of  permanence.*  Yet  this  and  this  only  is  justifiably  meant 
when  we  say  that  the  line-element  ds2  is  invariant  with 
respect  to  any  transformations  whatever.  If  the  relativist 
sees  anything  more  in  "the  invariance  of  ds",  namely  that 
ds  is  something  belonging  to  a  pair  of  world-points  (*,  and 
jct+ dxt)  inherent  in  that  pair  independently  of  the  choice  of 
a  reference  system,  it  is  what  he  puts  into  it  at  a  later  stage 
by  ascribing  to  it  certain  physical  properties,  or  by  inter- 
preting it  physically  in  certain  ways.  The  meaning  of  these 
remarks  will  gradually  become  more  intelligible. 

Before  passing  on  to  the  two  cardinal  virtues  conferred 
upon  the  line-element,  one  more  mathematical  remark  about 
it  may  not  be  out  of  place  just  here.  Suppose  the  line- 
element  (5)  is  actually  given  with  some  determined  and  more 
or  less  complicated  functions  as  the  glK.  By  trying,  in  succes- 
sion, other  and  other  new  variables  xtf  we  would  arrive  at  a 
great  variety  of  new  forms  of  functions  glKf.  The  natural 
question  arises:  Are  there  not  among  all  these  sets  of  co- 
ordinates just  such  as  would  convert  (5),  throughout  the 
world  or  a  finite  world-domain,  into  a  galilean  line-element, 
i.e.,  one  with  constant  coefficients?  The  answer  is,  in  general, 
in  the  negative.  A  given  form  ds2  =  gtK  dxt  dxK  is  equivalent,  that 
is  to  say,  can  be  reduced  by  holonomous  transformations,  to 
a  form  with  constant  coefficients  and  thus  also  to  the  galilean 
line-elementf  when  and  only  when  certain  differential 
expressions  formed  of  the  gw,  their  first  and  second  deriva- 
tives, all  vanish.*  These  expressions,  of  which  more  will  be 

*Notice  that  the  case  is  different  in  special  relativity,  where  we  require 
the  form  to  reappear  with  all  its  original  coefficients,  three  — 1,  and  one+1. 

fThe  circumstance  that  three  of  the  coefficients  of  this  form  are  negative 
and  one  positive  imposes  on  the  original  g  dx  dx  to  be  thus  transformable 
certain  further  conditions  in  connection  with  the  so-called  'law  of  (alge- 
braic) inertia',  due  to  Sylvester. 

*The  restriction  to '  holonomous  transformations '  is  of  prime  importance. 
For  by  means  of  non-holonomous  or  non-integrable  relations,  such  as 
every  g^  dxt  dxK  can  be  transformed  into  a  quadratic  differential  form  with 
constant  coefficients. 


RIEMANN'S  SYMBOLS  17 

said  in  the  sequel,  are  known  in  general  differential 
geometry  as  Riemann's  four-index  symbols.  Of  these 
symbols  there  are  in  the  case  of  any  number  n  of  dimensions, 

—  «2(w2  — 1)  linearly  independent  ones.     Thus  an  ordinary, 

two-dimensional,  surface  has  but  one  Riemann  symbol  and 
this  is  its  Gaussian  curvature,  multiplied  by  gng22— gi22,  the 
determinant  of  the  glK.  Any  three-dimensional  manifold  has 
six,  and  our,  or  rather  Einstein-Minkowski's  world  has  as 
many  as  twenty  linearly  independent  Riemann  symbols. 
Thus  any  finite  domain  of  the  world  is  equivalent  to  a  galilean 
domain  when  and  only  when  all  these  twenty  symbols  vanish 
in  that  domain,  i.e.,  when  the  ten  different  gllc  satisfy  within  it 
a  system  of  twenty  partial  differential  equations  of  the  second 
order.  (It  will  be  useful  to  keep  in  mind  the  last  italics.) 
By  what  has  just  been  said  it  is  manifest  that  if  all  the  Rie- 
mann symbols  vanish  in  one  system  of  coordinates  xit  they 
will  vanish  also  in  any  other  xj  obtained  from  the  former  by 
any  holonomous  transformations  whatever. 

But  enough  has  for  the  present  been  said  on  the  symbols 
of  that  great  geometer.  Later  on  they  will  be  seen  to  play 
an  all-important  role  in  Einstein's  gravitation  theory. 

It  is  now  time  to  return  to  the  physical  aspect  of  our 
subject. 

7.  Having  assumed,  after  Einstein,  that  special  relativity 
holds  for  every  infinitesimal  domain,  or  that  the  world  is 
elementally  galilean,  we  wrote  down  the  simple  form  (3)  in 
local  coordinates  ut.  Then,  passing  to  any  coordinates  xt  by 
means  of  the  non-holonomous  relations  (4)  we  obtained  for 
the  line-element  of  the  world  the  general  quadratic  differential 
form  (5),  with  variable  coefficients  g^,  functions  of  the  xt. 

But  what  is  the  physical  meaning  of  this  general  ds  with 
all  its  ten  different  gllc?  What  are  they  to  represent  physically? 
The  answer  is  that  we  are  still  to  a  certain  extent  the  masters 
of  the  situation,  and  can  make  them  have  that  physical 
meaning  which  we  will  put  into  them.  For  thus  far  we  know 
only  the  physical  meaning  of  the  galilean  element  belonging 
to  a  world  U,  and  that  (in  virtue  of  an  assumption)  the  world 


18  RELATIVITY  AND  GRAVITATION 

W  as  a  seat  of  or  deformed  by  gravitation  is  galilean  in  its 
elements,  or  that  at  each  of  its  points  a  £/-world  tangential  to 
it  can  be  constructed. 

At  this  stage  then  we  are  entitled  only  to  say  that  (since 
W  without  gravitation  is  £/,  and  since  to  U  belong  the  constant 
coefficients  ~glK)  the  essential  differences*  in  the  coefficients  of 
the  two  worlds,  g^  and  ~gtK,  are  due  to,  or  better,  are  somehow 
connected  with  gravitation.  But  exactly  how,  we  cannot, 
thus  far,  say.  For  our  position  is  somewhat  like  this :  Suppose 
we  know  that  a  surface  cr,  which  is  not  a  plane  as  a  whole,  is 
elementally  flat  and  thus  has  a  tangential  plane  TT  at  each  of 
its  points.  Suppose  further  we  know  the  physical  properties 
of  certain  lines  (straights,  or  circles,  etc.)  drawn  on  any  TT. 
Does  this  alone  enable  us  to  say  what  the  physical  properties 
of  similarly  defined  lines  will  be  when  drawn  on  o-?  Clearly 
not.  For  the  7r-lines  have  but  a  single  point  of  contact  with 
cr,  and  that  only  of  the  first  order,  and  deviate  from  the 
surface  or  become  extra-a  beings  all  around  the  point  of 
contact. 

Now,  in  the  case  of  space- time,  we  fixed  the  physical 
meaning  of  the  line-element  of  the  £/-world  by  declaring  its 
minimal  lines,  ds  =  Q,  to  be  the  law  of  light  propagation,  and 
its  geodesies,  dfds  =  0,  to  represent  the  motion  of  free  particles. 
Does  this,  and  the  existence  of  a  tangential  U  at  every  point 
of  the  actual  world  W,  entitle  us  to  assert  that  the  minimal 
lines  and  the  geodesies  of  W  will  again  represent  the  optical 
and  the  mechanical  laws  in  this  world?  This  is  by  no  means 
a  superfluous  question.  For  the  auxiliary  tangential  world 
U  leaves  the  actual  world  beyond  the  point  of  contact  and 
becomes  at  once  fictitious  or  extra-mundane,  so  to  speak. 

Now,  the  minimal  lines  of  £/,f  defined  by  a  differential 
equation  of  the  first  order,  are  also,  at  P,  minimal  lines  of  W, 
so  that  at  least  the  starting  elements  of  these  lines  are  identical. 
At  the  next  element  the  role  of  U  is  taken  over  by  another 

*i.e.,  those,  at  least,  which  cannot  be  abolished  by  holonomous  co- 
ordinate transformations. 

fWhich  fill  out  only  a  conic  hypersurface  (of  three  dimensions)  with  the 
contact  point  P  as  apex. 


GEODESICS  AND  LAW  OF  MOTION  19 

galilean  world;  yet  the  reasoning  can  be  repeated,  so  that  we 
can  say  that  every  element  of  a  minimal  line  of  W  represents 
light  propagation,  and  thence  deduce  that  such  a  HMine 
possesses  also  as  a  whole  the  same  physical  property.  But 
the  position  is  altogether  different  with  the  geodesies.  For 
these  world-lines  are  defined  by  differential  equations  of  the 
second  order.*  so  that  the  mere  contact  of  U  and  W  (being  of 
the  first  order)  does  not  at  all  entitle  us  to  transfer  any 
properties  of  the  geodesies  of  U  upon  those  of  W,  not  even 
at  their  very  starting  point  P. 

If,  however,  the  said  physical  property  of  the  PF-geodesics 
does  not  follow  logically  from  the  previous  assumptions,  yet 
we  are  free  to  introduce  it  as  a  further  explicit  assumption. 
In  fact,  while  thus  generalizing  the  physical  significance  of  the 
geodesies  Einstein  is  well  aware  that  this  is  a  new  assumption,! 
although  one  that  easily  suggests  itself.  Nor  is  there  any 
inconsistency  in  thus  transfering  a  property  from  the  galilean 
to  the  more  general  world-geodesies.  For,  as  we  shall  see 
later  on,  the  developed  form  of  the  equations  of  the  geodesies 
contains  only  the  g^  and  their  first  derivatives  with  respect 
to  the  xtj  whereas  the  conditions  characterizing  a  world  as 
galilean  (the  vanishing  of  the  Riemann  symbols)  are  equations 
between  the  glKJ  their  first  and  second  derivatives,  and  there 
are  no  relations  at  all  between  the  glK  and  their  first  derivatives 
alone. 

But  even  with  this  new  assumption,  the  total  number  of 
assumptions  of  Einstein's  theory  is  remarkably  small.  And 
as  to  the  advisability  of  making  the  one  just  discussed,  we 
may  say  that  Einstein's  theory  owes  to  it  the  greater  part  of 
its  power. 

The  property  of  the  geodesies  being  thus  assumed,  and 
that  belonging  to  the  minimal  lines  being  deducible  from 
what  preceded,  we  are  now  in  the  positibn  to  sum  up  definitely 
and  very  concisely,  if  not  the  whole,  yet  the  most  fundamental 
part  of  Einstein's  theory.  For  this  purpose  we  have  only  to 

*A  geodesic  issues  from  P  in  every  direction  whatever  in   the  four- 
manifolds  U  and  W. 

fA.  Einstein,  loc.  cit.t  p.  802. 


20  RELATIVITY  AND  GRAVITATION 

repeat  the  previous  statements  I.  and  II.  without  their 
restrictions,  replacing  the  galilean  ds  by  the  general  one  and 
adding  a  few  explanatory  words:  Thus: 

The  world-line  element,  in  any  system  of  coordinates,  and 
whether  gravitation  be  absent  or  present,  is  given  by 

ds*  =  glKdxtdxK,  (10) 

where  g«  =  g«  are  some  functions  of,  in  general,  all  the  Jour 
coordinates,  but  of  these  alone.  If  these  ten  functions  be  given, 
all  metrical  properties*  of  the  world  are  determined,  and  among 
these  its  minimal  lines, 

ds  =  0,  (I) 

and  its  geodesies, 

0.  (II) 


The  physical  significance  of  these  world-lines  is  that  the  former 
represent  propagation  of  light  in  vacuo,  and  the  latter  the  motion 
of  a  free  particle. 

By  a  'free'  particle  is  meant  one  which,  having  received 
any  initial  impulse  is  left  to  its  own  fate,  whether  in  absence 
or  in  proximity  of  other  lumps  of  matter  (absence  or  presence 
of  'gravitation'),  but  not  colliding  with  them,  and  in  absence 
of,  or  better  not  immersed  in,  an  electromagnetic  field.  One 
strives  in  vain  to  enumerate  all  the  attributes  of  a  concept 
which  can  become  clear  only  a  posteriori,  through  the  concrete 
applications  of  the  theory.  Suffice  it  to  say  that  '  free  particle  ' 
may  as  well  stand  for  a  projectile,  in  vacuo,  or  a  planet 
circling  around  the  sun.  Their  laws  of  motion  are  given  by 
the  corresponding  world-geodesies.  The  developed  form  of 
the  equations  of  the  geodesies,  as  well  as  of  light  propagation, 
will  be  given  later  on. 

Since  the  g^  are  to  determine,  through  (II),  the  fall  of 
projectiles  and  the  motion  of  celestial  bodies,  it  is  scarcely 
necessary  to  repeat  that  they  are  intimately  connected  with 
gravitation.  These  ten  coefficients  will  replace  the  unique 
scalar  potential  of  newtonian  mechanics.  They  will  influence 

*Apart  from  some  properties  of  the  world  as  a  whole,  —  of  which  more 
later  on. 


FREE  PARTICLES  AND  LIGHT  21 

also,  through  (I),  the  course  of  light  in  interplanetary  and 
interstellar  spaces,  and  finally,  by  their  very  appearance  in 
the  line-element,  they  will  mould  the  geo-  and  chrono-metrical 
properties  of  our  world.  These  latter  properties  thus  appear 
intimately  entangled  with  gravitation  and  optics. 

It  remains  to  explain  how  these  all-powerful  coefficients 
are,  in  their  turn,  determined  in  terms  of  other  things  such 
as  the  density  of  'matter'.  This  is  the  office  of  Einstein's 
'field-equations'  which  will  occupy  our  attention  in  the 
sequel. 


CHAPTER  II. 

The  General  Relativity  Principle.    Minimal  Lines  and 

Geodesies.    Examples.     Newton's  Equations  of 

Motion  as  an  Approximation. 


8.  Most  readers  will  perhaps  be  surprised  to  find  in  the 
first  chapter  almost  no  mention  of  the  general  principle  of 
relativity  which  claims  equal  rights  for  all  systems  of  co- 
ordinates, and  which  in  all  publications  on  our  subject  is 
given  the  most  prominent  place.  Instead  of  this  we  insisted 
on  the  general  form  of  the  line-element  (10),  on  the  null-lines 
and  the  geodesies  of  the  world  metrically  determined  by  that 
line-element,  and  still  more  upon  the  physical  meaning  of 
these  two  kinds  of  world-lines  as  representing  light  propa- 
gation and  the  motion  of  free  particles. 

The  reason  for  adopting  this  plan  is  that,  as  far  as  I  can 
see,  these  things  are  most  important  from  the  physical  point 
of  view,  nay,  they  are  perhaps*  the  only  relevant  constituents 
of  the  new  theory  looked  upon  as  a  physical  theory.  This  is 
particularly  true  of  the  optical  and  mechanical  meaning 
attributed  to  the  said  two  kinds  of  lines,  thus  giving  what  the 
logicians  call  a  concrete  representation  of  what  otherwise 
would  be  only  a  purely  mathematical  or  logical  science,  an 
abstract  geometry  of  a  manifold  of  four  dimensions  deter- 
mined by  that  quadratic  differential  form.  It  is  exactly  this 
physical  interpretation  which  invests  the  theory  with  the 
power  of  making  statements  of  a  phenomenal  content,  of 
predicting  the  course  of  observable  events.  On  the  other 
hand,  the  much  extolled  Principle  of  General  Relativity 
which,  in  Einstein's  wording, f  requires 
The  general  laws  of  Nature  to  be  expressed  by  equations  valid 


*Apart  from  '  the  field  equations ',  yet  to  come. 
\Loc.  cit.t  p.  776. 

22 


GENERAL  RELATIVITY  PRINCIPLE  23 

in  all  coordinate  systems,  i.e.,  covariant  with  respect  to  any 
substitutions  whatever  (generally  covarianl), 
is  by  itself  powerless  either  to  predict  or  to  exclude  anything 
which  has  a  phenomenal  content.  For  whatever  we  already 
know  or  will  learn  to  know  about  the  ways  of  Nature,  pro- 
vided always  it  has  some  phenomenal  contents  (and  is  not  a 
merely  formal  proposition),  should  always  be  expressible  in 
a  manner  independent  of  the  auxiliaries  used  for  its  descrip- 
tion. In  other  words,  the  mere  requirement  of  general 
covariance  does  not  exclude  any  phenomena  or  any  laws  of 
Nature,  but  only  certain  ways  of  expressing  them.  It  does 
not  at  all  prescribe  the  course  of  Nature  but  the  form  of  the 
laws  constructed  by  the  naturalist  (mathematical  physicist 
or  astronomer)  who  is  about  to  describe  it.  The  fact  that 
some  phenomenal  qualities  are  technically  (with  our  inherited 
mathematical  apparatus)  much  more  difficult  to  put  into  a 
generally  covariant  form  than  some  others  does  not  in  the 
least  change  the  position. 

To  make  my  meaning  plain,  let  us  take  the  case  of  plane- 
tary motion.  For  the  sake  of  simplicity  let  there  be  but  a 
single  planet  revolving  around  the  sun.  It  is  well-known  that 
according  to  Newton  the  orbit  of  the  planet  should  be  a 
conic  section,  say  an  ellipse  with  fixed  perihelion.*  It  is,  in 
our  days,  almost  equally  well  known  that  according  to 
Einstein's  theory  the  perihelion  should  move,  progressively, 
showing  a  shift  at  the  completion  of  each  of  its  periods.  And 
so  it  does,  at  least  to  judge  from  Mercury's  behaviour.  At 
the  same  time  Einstein's  equations  are  generally  covariant, 
while  Newton's  'law'  or  Laplace- Poisson's  equations  are  not.f 
What  of  this?  Does  it  mean  that  fixed  perihelia  are  excluded 
or  prohibited  by  the  principle  of  general  covariance?  Cer- 
tainly not.  Provided  that  'fixed  perihelion'  and  'moving 
perihelion'  have,  each,  a  phenomenal  content,  and  this  they 
do,  both  kinds  of  planetary  behaviour  should  be  expressible 
in  a  generally  covariant  form.  Newton's  inverse  square  law 
and  his  equations  of  motion  certainly  do  not  express  it  so, 

*Fixed,  that  is,  relatively  to  the  stars. 

fNot  even  with  respect  to  the  special  or  the  Lorentz  transformation. 


24  RELATIVITY  AND  GRAVITATION 

and  it  may  be  difficult  to  find  a  covariant  expression  for  a 
strictly  keplerian  behaviour.  But  if  it  were  urgently  needed, 
some  powerful  mathematician  would,  no  doubt,  succeed  in 
constructing  it.  If,  as  actually  is  the  case,  Einstein's  theory 
excludes  a  fixed  perihelion,  and  other  newtonian  features,  it 
does  this  not  in  virtue  of  the  said  principle  alone  (nor  even  in 
part),  but  pre-eminently  owing  to  the  physical  meaning 
ascribed  to  the  world-geodesies,  and  to  the  choice  of  his  field 
equations  which  again  are  physically  relevant  since  they 
determine  the  giK  influencing  essentially  the  form  of  those 
world-lines.  That  the  principle  of  general  relativity  turned 
out  to  be  helpful  in  guessing  new  laws  (by  limiting  the  choice 
of  formulae)  is  an  altogether  different  matter.  It  may  prove 
an  even  more  successful  guide  in  the  future. f  But  here  its 
role  ends, — always  taking  the  Principle  only  as  a  mathematical 
requirement  of  general  covariance  of  equations.  And  so  it 
is  at  any  rate  enunciated  (and  interpreted,  cf,  p.  776,  loc.  ciL) 
by  Einstein  himself,  although  some  of  his  exponents  put  into 
it  a  physical  meaning.  In  fact,  as  we  shall  see  later  on,  the 
sameness  of  form  of  the  equations  (of  motion,  say)  in  two 
reference  systems,  as  in  a  smoothly  rolling  and  a  vehemently 
jerked  car,  does  not  at  all  mean  sameness  of  phenomenal 
behaviour  for  the  passengers  of  these  two  vehicles. 

So  much  in  explanation  of  the  absence  of  the  general 
principle  of  relativity  in  all  our  preceding  deductions. 

It  will  be  noticed,  however,  that  although  no  explicit 
mention  of  this  principle  has  been  made  in  Chapter  I,  yet  the 
fundamental  laws  (I)  and  (II)  there  given  do  satisfy  this 
principle.  In  fact,  both  the  null-lines  and  the  geodesies  of 
the  world  were  defined  without  the  aid  of  any  reference 
system.  And  as  to  the  line-element  itself,  its  invariance  was 
seen  to  be  automatic. 

Thus,  in  what  precedes  we  have,  without  insisting  upon 
it,  been  faithful  to  the  formal  principle  of  general  relativity. 
Nor  is  it  our  intention  to  depart  from  it  in  what  will  follow. 

fOr  it  may  become  sterile  to-morrow,  as  is  the  fate  of  almost  all  our 
Principles. 


LIGHT  VELOCITY  25 

As  was  already  mentioned  at  the  close  of  the  first  chapter, 
to  make  the  exposition  of  the  fundamental  part  of  Einstein's 
theory  complete,  it  remains  to  add  to  (10),  (I),  (II),  together 
with  their  optical  and  mechanical  meaning,  a  set  of  equations 
determining  the  ten  coefficients  g^  of  the  quadratic  form. 
But  before  passing  to  these  differential  equations,  Einstein's 
field-equations,  it  will  be  well  to  discuss  somewhat  more  and 
to  develop  those  already  given.  Some  explanations  and 
examples  concerning  the  transformation  of  coordinates  will 
also  be  helpful  at  this  stage. 

9.  First,  concerning  the  law  of  propagation  of  light 
(in  vacuo),  to  obtain  its  developed  form  it  is  enough  to  sub- 
stitute the  line-element  (10)  into  the  equation  (I)  of  the 
minimal  lines.  Thus  the  fundamental  optical  law  will  be 

g^dx.dx^O.  (11) 

It  gives  the  velocity  of  light  for  every  direction  of  the  ray,  i.e.,  of 
the  infinitesimal  space-vector  dx\,  dx2,  dxs,  if  dx^/c  be  the 
time  element  of  the  reference  system.  In  general  the  light 
velocity  will  differ  from  c  and  have  different  values  at  different 
world  points  and  for  different  directions  of  the  ray. 

This  "light  velocity"  which  has  nothing  intrinsic  about 
it  is  to  be  distinguished  from  the  local  velocity  of  light  (that 
corresponding  to  a  local,  galilean  system  of  coordinates) 
which  is  the  same  for  all  directions.  To  avoid  confusion  the 
former  may  be  called  the  system-velocity  of  light  or,  according 
to  some  authors,  the  'coordinate  velocity'  of  light.  It  is  a 
kind  of  velocity  estimated  from  a  distant  standpoint.  If  we 
write  it,  in  a  given  reference  system, 

da  _      da 
~Jt~    Cdx~t' 

the  very  concept  of  such  a  light  velocity,  whose  value  is  to 
be  derived  from  (11),  presupposes  that  'the  length'  da  of  the 
infinitesimal  space-vector  dxit  dx^,  dx3  has  been  defined  in 
some  way  for  that  system  in  terms  of  these  differentials  and 
the  coefficients  gllc.  We  shall  have  the  best  opportunity  of 


26  RELATIVITY  AND  GRAVITATION 

explaining  how  this  is  done  technically  in  deducing  physical 
results,  when  we  come  to  speak  of  the  bending  of  rays  of  light 
around  a  massive  body  such  as  the  sun.  Then  also  the 
question  will  be  mentioned  under  what  circumstances  the  law 
of  Fermat,  giving  the  shape  of  the  rays,  is  applicable. 

In  the  meantime  it  is  advisable  to  look  upon  (11)  as  the 
equation  of  the  infinitesimal  wave  surface  at  the  instant  t-\-  dt 
corresponding  to  a  light  disturbance  started  at  Xi,  xz,  xz  at  the 
instant  /,  the  differential  dx^  being  treated  as  a  constant 
parameter.  From  the  local  standpoint  this  surface  is,  of 
course,  a  sphere,  but  from  the  distant  (or  system-)  standpoint 
it  may  have  a  variety  of  more  complicated  shapes.  It  would, 
perhaps,  be  rash  to  say  that  it  will  be  a  quadric.  But,  being 
locally  closed,  it  may  also  be  expected  to  be  a  closed  surface 
from  the  system-point  of  view. 

10.  Next  for  the  geodesies  of  the  world.  The  developed 
form  of  their  differential  equations  is  easily  derived  from  their 
original  definition  (II), 


As  in  the  case  of  a  galilean  world,  let  u  be  any  parameter,  and 
let  dots  stand  for  derivatives  with  respect  to  it.    Then 

fds  .du  =  Q, 
where,  in  the  most  general  case, 

s2  =  £„*.*«.  (12) 

The  variation  of  s  can  be  written 

8s  =  —  BXl+  —  dxt, 
dxt  dxt 

to  be  summed  over  t  =  1  to  4.    Thus,  by  partial  integration  of 
the  second  terms,  the  limits  of  the  integral  being  fixed, 

d,_  /j9s  \  _  ds_  =  0, 
du  V  dxt  /        dxt 

and  by  (12),  with  5  itself  taken  for  u, 


GEODESICS 


27 


\  _  ±  dga(i   dxa 
/        "  d#      Js 


or 


dx 
ds 

^_l_  dg^^t  _  i      ^^        =  Q 
ds2        d#x  ds  ds  dxt  ds   ds 

Introducing  the  expressions,  known  as  Christoffel's  symbols, 

fan  =   /a^a    as,  _  ^  =  r^n 

LTJ        V^       ajca      dxy/      LTJ 

we  can  condense  the  last  set  of  equations  into 


dxa  dxp  _ 

= 


These  are  four  linear  equations  for  the  four  d?xK/dsz.  Let  us 
solve  them  for  these  derivatives.  Denoting  the  second  term 
by  alt  and  writing  g  for  the  determinant  of  the  gl(e,  we  shall  have 


0,  etc. 


or,  if  g"c  =  gKt  be  the  minor  of  g,  corresponding  to  glK,  divided  by 
g  itself, 

-f-  dig11  H~  Q-zg12  ~f~  &3g13  ~f~  #4&14  ~  0,  etc., 


i.e., 


t  ap  =  Q 

ds2        '    L  «  J  ds  ds  ~ 

Here  we  will  write,  after  Christoffel, 


-3 


(14) 


28  RELATIVITY  AND  GRAVITATION 

Thus,  ultimately,  the  differential  equations  of  the  geodesies 
or  the  equations  of  motion  of  a  free  particle  will  be,  in  any  system 
of  coordinates, 


**        (o01^   dx, 
ds*       I  O  ds     ds 


These  are  four  equations.    But  since  we  have,  identically, 

dxt    dxK 

&IK  -        '  -     :=      1> 

ds    ds 

one  of  these  equations  of  motion  is  a  consequence  of  the 
remaining  three,  a  feature  already  familiar  to  the  reader  from 
special  relativistic  mechanics.  Since  these  differential  equa- 
tions are  only  the  developed  form  of  dfds  =  0,  they  will  mani- 
festly be  generally  covariant,  that  is  to  say,  in  any  new 
coordinates  xj  the  equations  (15)  will  be 


d2xj    ,    |  a/3)  ' 

^r  +  W 


ds     ds 


If  the  coefficients  glK  are  all  constant,  all   the  Christoffel 
symbols  \      f    vanish    and     the    equations    (15)    reduce    to 

d9xjds*  =  0,  which  represent  uniform  rectilinear  motion. 
And  since  the  general  equations  (15)  represent  the  motion  of 
a  free  particle  in  any  gravitational  field  and  in  any  system,  the 


symbols  <      >,  built  up  of  the  giK  and  their  first  derivatives, 

can  be  said  to  express  the  deviation  of  the  motion  from 
uniformity  due  to  gravitation,  and  partly  due  to  the  peculiari- 
ties of  the  system  of  reference.  In  view  of  this  property,  and 
disregarding  any  distinction  between  gravitation  proper  and 
the  effects  of  the  choice  of  the  coordinate  system,!  Einstein 


*This  form  of  the  equations  of  a  geodesic  of  a  manifold,  of  any  number 
of  dimensions,  has  been  used  by  geometers  for  a  long  time.  See,  for 
instance,  L.  Bianchi's  Geometria  differ enziale,  vol.  I,  Pisa  1902,  p.  334. 

fOr  between  permanent  acceleration  fields  and  such  that  can  be  trans- 
formed away. 


CHRISTOFFEL  SYMBOLS  29 

proposes  to  call  these  Christoffel  symbols  '  the  components  of 
the  gravitational  field'. 

Notice,  however,  that  if  all  <      >  vanish  in  one  system 

of  reference  they  do  not  necessarily  vanish  in  other  systems* 
(even  if  obtained  from  the  former  by  holonomous  transforma- 
tions). In  view  of  this  circumstance  the  name  proposed  by 
Einstein  seems  utterly  inappropriate  and  misleading,  even 
if  one  agreed  not  to  distinguish  between  permanent  fields 
and  such  that  can  holonomously  be  transformed  away,  as  for 
instance  the  'centrifugal  force'. 

10a.   In  fact,  consider  for  example  the  galilean  line-element 
in  three  dimensions,  i.e.,  for  $  =  const.  =  7r/2, 


taking  c/,  r,  0  as  x4,  #1,  #2  respectively.  Calculate  the  corres- 
ponding Christoffel  symbols.  Since  gn=  —  1,  £22=  —  f2,  £44=  1, 
and  all  other  gtK  vanish,  we  have,  for  instance,  the  non-vanish- 
ing symbol 

(22] 

But  who  would  call  it  a  '  component  of  the  gravitational  field ' ? 
This  case  is  a  particularly  drastic  one,  for  the  world-geodesies 
corresponding  to  our  line-element  do  represent  uniform  recti- 
linear motion.  The  appearance  of  non-vanishing  Christoffel 
symbols  is  simply  due  to  the  use  of  polar  instead  of  cartesian 
co-ordinates. 

In  short,  gravitation  certainly  contributes  to  the  Chris- 
toffel symbols,  but  so  does  also  a  mere  transformation  of 
space-coordinates,  although  it  has  nothing  whatever  in 
common  with  'gravitation'  of  the  permanent  or  the  non- 
permanent  kind.  This  criticism  does  not  in  the  least  diminish 
the  value  of  the  general  equations  of  motion  (15).  It  is  given 
here  only  to  prevent  misconceptions  which  have  seemed 
particularly  likely  in  the  case  of  beginners. 

*In  the  terminology  of  the  tensor  calculus,  to  be  explained  later  on,  the 
Christoffel  symbols  are  not  the  components  of  a  tensor. 


30  RELATIVITY  AND  GRAVITATION 

lOb.  Let  us  take  yet  another  simple  example,  this  time 
not  for  the  sake  of  criticism  but  because  of  its  instructiveness. 
Consider  the  line-element  arising  from  the  galilean  one, 
just  quoted, 

(£')          ds2  =  dx'S  -  dr'2  -  /  W2, 
by  the  transformation 

0'  =  8+<axt,      x't  =  xt,      r'  =  r,  (16) 

that  is  to  say,  the  line-element 

(S)         ds2  =  (1  -  r2co2)dx42  -  dr*  -  rW  -  2corWd*4. 
In  this  case,  taking  r,  6  as  Xi,  x2  respectively,  the  non-  vanishing 
g«  are 

gu  =  —  1  ,  g22  =  —  r2,  g24  =  —  cor2,  g44  =  1  —  wV2. 
From  these  we  derive,  by  (13),  as  the  only  surviving  Chris- 
toffel  symbols, 

-.  ra  —  .  m  —  .  KI-*. 


Next.  we  have,  the  determinant  of  the  form  (5), 

g 

and 


while  all  other  glK  vanish.    Thus  we  find,  by  (14),  as  the  only 
non-vanishing  symbols, 

12 


14  22 


/44\  J24\ 

-r>i-«*r>  i=-^ 


again  seven  in  number.  Substituting  these  Christoffel 
symbols  into  (15),  with  i  =  l,  2,  4  (for  r,  0,  xi  =  ct),  we  have 
the  equations  of  the  world-geodesies,  i.e.,  the  equations  of 
motion  of  a  free  particle  in  the  system  S, 


ROTATING  SYSTEM 


31 


r   = 


(l-2coV)(0+co*4) 


(17) 


In 


where  the  dots  stand  for  derivatives  with  respect  to  s. 
virtue  of  the  identical  equation  s=  1,  i.e., 

(l-r2co2)*42-  r2—  r2  02-2cor20*4=  1,  (18) 

one,  say  the  third  of  (17),  should  be  a  consequence  of  the 
remaining  two.*  Thus,  the  proper  equations  of  motion  in  the  S- 
system  being  the  first  two  alone,  we  can  use  (18)  to  eliminate 
from  them  #4,  and  to  replace  d/ds  by  d/dt. 

In  the  first  place,  to  see  the  approximate  meaning  of  these 
equations  of  motion,  consider  the  case  of  small  velocities 
dr/dtj  rdd/dt  (as  compared  with  c),  and  of  small  values  of  cor. 
[Notice  that,  by  (16),  co  =  coc  is  an  angular  velocity,  in  its 
dimensions  at  least,  so  that  cor  =  ur/c  is  a  pure  number.]  Thus 
ds=^dx^  =  cdt,  £4=Fl,  and  the  approximate  equations  of  motion 
of  a  free  particle  in  5  are 

drz 


dt* 


=  -2 


dr_ 
dt 


j 

+ 


dt 


In    Cartesians,    #  =  rcos0,    y  = 
identical  with 


dy 


/,    these    equations    are 


(b) 


d*y        A      0  -    dx 

=  coy  —  2,  co 

dt*  dt 


The  reader  will  recognize  at  once  in  the  right  hand  member  of 
equation  (a)  or  in  the  first  terms  of  (b)  the  purely  radial 
centrifugal  acceleration  (or  'force'  per  unit  mass),  provided, 

*The  verification  may  be  left  to  the  reader  as  an  exercise. 


32  RELATIVITY  AND  GRAVITATION 

of  course,  that  he  is  at  all  willing  to  interpret  o>,  in  accordance 
with  the  transformation  0'  =  0-fo>/,  as  the  angular  velocity  of 
the  system  S  (say,  plane  disc)  relatively  to  the  galilean  Sf. 
The  second  terms  of  (b)  express  then  the  Coriolis  acceleration. 

If  we  so  desire  we  may,  with  Einstein,  reckon  these  accelerations  to  the 
gravitational  ones,  especially  if  we  are  confined  to  the  (rotating)  system  S. 
The  centrifugal  acceleration,  at  least,  is  radial,  though  away  from  the 
origin.  The  Coriolis  acceleration,  however,  is  perpendicular  to  the 
velocity  and,  therefore,  generally  oblique.  Certainly  we  have  in  (17)  a  field 
of  acceleration,  but  the  only  feature  this  has  in  common  with  a  gravita- 
tional field  is  that  all  bodies  placed  in  it  will  behave  alike.  But  unlike 
gravitational  fields  they  cannot  be  deduced  from  the  distribution  of  matter. 
Yet  Einstein  would  not  like  to  have  us  distinguish  them  from  gravita- 

1 22  i       1 24 1      1 44 1 
tional  fields.    If  so,  then  1 1  f  •  i  |  f  »  "i  1  f  contribute  to  the  centrifugal, 

(     )         (     ) 

and    I  2  (  '     I  9  (    t0  the  Coriolis  field-      But  until  we  are  told  now  to 

derive  these  non-permanent  'fields'  as  gravitational  effects  of  all  the 
masses  of  the  universe  turning  around  S*  all  this  will  be  an  idle  question 
of  pure  nomenclature.  We  may  leave  it  here  for  the  present. 

In  the  second  place,  returning  to  the  rigorous  equations 
(17),  consider  a  particle,  placed  (by  an  5-inhabitant)  at  any 
point  r0,  60  of  the  disc  S  and  left  there,  at  the  instant  /  =  0,  to 
its  own  fate.  If  it  is  nailed  down  it  will,  of  course,  remain 
there  for  ever,  being  simply  part  of  this  reference  system. 
But  let  it  be  a  free  particle  from  /  =  0  onwards.  In  short,  let 
r=  0  =  0,  for  /  =  0.  Then,  by  (17),  we  shall  have,  for  that 
instant,  6  =0  so  that  the  particle  will  not  evince  any  tendency 
of  moving  transversally,  and 


ds* 


d     /.      dr\ 

=  — -1*4  — -  I  = 
dx4  \       dt  / 


By  (18),  £42  =  (1  —  r2cu2),  and  since  r0  =  0,  the  last  equation  will 
become,  rigorously,  and  always  for  /  =  0, 


dt2 


*This  was  tried  by  H.  Thirring  but  not  very  successfully. 


ROTATING  SYSTEM  33 

In  fine,  our  particle  will  initially  experience  the  familiar 
centrifugal  acceleration.*  It  will  fly  off,  for  an  S'-observer 
at  a  (straight)  tangent,  but  from  the  5-standpoint  at  a 
spiral-shaped  orbit. 

This  is  perhaps  the  clearest  way  of  stating  the  relation 
of  our  system  5  to  the  galilean  S'.  The  reader  need  not, 
however,  think  of  5  at  this  stage  as  a  material  rigid  disc 
rotating  uniformly  with  respect  to  the  fixed  stars,  although 
a  uniform  rotation  is  just  one  of  the  possible  motions  of  a 
relativistically  rigid  body  (Born,  Herglotz).  Notwithstanding 
that  S  was  called,  in  passing,  a  disc,  it  will  be  safer  to  treat 
it  here  simply  as  a  system  derived  from  S'  by  the  trans- 
formation (16)  with  co  as  constant. 

As  to  the  orbit  of  a  free  particle  relatively  to  5,  its  equation 
could  be  derived,  not  without  some  trouble,  from  the  differ- 
ential equations  (17).  This,  however,  can  be  done  much 
easier  by  transforming  the  orbit  from  S'  to  5.  In  fact,  the 
former  being  a  galilean  system,  a  free  particle  describes  in  it, 
uniformly,  a  straight  line.  Its  equation  can  be  written 

r'  cos  6'  =  r0f  =  const., 

where  r0f  is  the  shortest  distance  of  the  straight  orbit  from 
the  origin.  Transformed  by  (16)  the  orbit  in  5  will  be 

!•  =  cos 

r 


and  since  v't'=Vr2  —  r02,  where  v'  is  the  constant  S'- velocity 
of  the  particle,  we  shall  have  ultimately,  as  the  orbit  of  a  free 
particle  in  5, 


^-V  --i    •          <19> 

v 


a 


*One  of  Einstein's  most  vigorous  exponents,  de  Sitter,  sees  herein  a 
particularly  extravagant  property  of  the  rotating  system.  Thus  in  Monthly 
Notices  of  the  Roy.  Astron.  Soc.,  vol.  77  (1916),  p.  176,  de  Sitter  says: 
'For  ro><!'  [and,  as  we  saw,  for  any  rco]  'it  is  a  physical  impossibility  for 
a  material  body  to  be  at  rest  in  the  system  B'  [our  S].  'This  shows  the 
irreality  of  the  coordinates',  etc.  But  such  is,  in  reality,  the  behaviour  of 
free  particles  in  a  system  rotating  relatively  to  the  stars,  independently  of 
any  theory. 


34  RELATIVITY  AND  GRAVITATION 

which  is  a  kind  of  spiral.  Notice  in  passing  that  between  any 
two  points  A,  B  of  the  disc  there  are  two  such  orbits,  one 
leading  from  A  to  B  and  the  other  from  B  to  A.  Thus  free 
motion  in  5  is  not  reversible.  This  holds  also  for  light  rays, 
for  which  v'  in  (19)  is  to  be  given  the  value  c.  Light  propaga- 
tion is  irreversible,  and  the  two  rays  AB  and  BA  enclose  a 
certain  area  having  the  shape  of  a  biconvex  lens.  But  this  by 
the  way  only. 

The  example  of  these  two  systems,  Sf  and  5,  was  here 
treated  at  some  length  in  order  to  acquaint  the  reader  with 
the  handling  of  the  geodesies  and  the  Christoffel  symbols. 
At  the  same  time,  however,  it  may  serve  as  a  good  illustration 
of  the  purely  formal  part  played  by  the  principle  of  general 
relativity  or  general  covariance.  In  fact,  although  the  equa- 
tions of  motion  of  free  particles  have  exactly  the  same  form, 
(15)  and  (15)  dashed,  in  the  two  systems,  yet  it  is  scarcely 
possible  to  imagine  a  more  different  phenomenal  behaviour 
of  free  particles  than  is  that  in  these  two  systems.  The  same 
remark  applies  to  the  light  equations,  g^' dxj  dxKf  —  0  in  S' 
and  glKdxtdxK  =  Q  in  5,  exhibiting  the  same  general  form,  but 
representing  entirely  different  systems  of  optics;  this  differ- 
ence goes  even  so  far  that,  while  in  S'  all  light  paths  are 
reversible,  in  5,  under  appropriate  conditions,  Brown  could 
see  Jones  without  being  visible  to  him,  though  both  were 
well  enough  illuminated. 

The  purpose  of  these  remarks  is  by  no  means  to  minimize 
the  heuristic  value  of  the  general  relativity  principle,  but  only 
to  show  its  purely  formal  nature.  Notice  that  the  case  of  the 
special  relativity  theory  was  altogether  different;  for,  though 
giving  privileges  only  to  a  certain  class  of  systems,  it 
claimed  at  least  for  all  of  them  not  only  a  formal  equality, 
but  an  equal  physical  behaviour. 

In  passing  from  Sf  to  S  the  Lorentz  contraction  was,  for 
the  sake  of  simplicity,  altogether  disregarded.  This  is  the 
reason  why  the  reader  was  warned  not  to  take  our  5  strictly 
as  a  rigid  body  rotating  in  S'  but  only  as  one  obtained  from 
S'  by  the  simple  mathematical  transformation  (16).  Yet 
even  with  the  said  neglect  the  abstract  5  can  at  least 


EQUATIONS  OF  MOTION  35 

approximately  stand  for  a  rigid  body,  such  as  the  earth  (any 
plane  r,  6  parallel  to  its  equator),  endowed  with  a  uniform 
spin  relatively  to  the  stars. 

11.  Leaving  these  simple  examples  let  us  once  more  return 
to  the  general  equations  of  motion  of  a  free  particle, 

*,  =  0  (15) 

in  order  to  see  what  form  they  assume  when  the  glK  differ  but 
little  from  the  galilean  coefficients  g«  and  when  Xi,  x2,  x$  are 
small  fractions,  that  is  to  say,  when  the  velocity  of  the  particle 
is  small  compared  with  that  of  light. 

If  the  galilean  line-element  is  written  in  Cartesians  we 

have  gn  =  £22  =  £33  =  —  1,  g44=l»  and 

gn=  -l+7ii,  etc.,  £44=  1+744, 


where  all  the  7  are  small  fractions  of  unity.  With  these 
values  of  the  gllc  we  could  compute  the  approximate  values 
of  the  Christoffel  symbols  appearing  in  (15),  and  thus  arrive 
at  the  required  equations.  But  it  is  simpler  to  return  to 
8fds  =  Q,  the  original  form  of  (15),  to  reduce  the  element  ds 
and  then  to  develop  this  form  afresh. 

Now,  if  dxi/cdt  =  fa,  etc.,  0i*+022+032  =  02,  the  line-element 
can  be  written 


(7ll012  +    .    •    •    +  733032 
+  7310301)  +  2(71401+    -    -   •    +73403)}. 

All  the  squares  and  products  of  the  0's  are  small  of  the  second 
order.    Thus,  up  to  the  third  order  we  have 


-p  +744  +  2  (0!7l4+    •    .    .    +  03734),  (22) 

and  the  equations  of  motion,  JdL  .  Jjc4  =  0,  will  be 


~i~)-  —  -o,  <=»•»•»• 


36  RELATIVITY  AND  GRAVITATION 

Now, 

dL         1  .    dL        1     "  1  5744     ,    . 


and  if  the  squares  and  the  products  of  the  y's  and  their 
derivatives  be  neglected,  we  can  put  L==l  in  the  denominators. 
Thus  the  equations  of  motion  will  become 

d         f  fl\  1        ^744        ,„        #741        .a        #742         ,    a       #743 

-  -    (74*-  ft)    =    2    —  --  h&  —  --  H&—  I  ---  HA  -I  -    ' 
d#4  dtfj  6^  5^,-  5oCt- 

or,  developing  the  first  term  and  remembering  that  the  ylK 
differ  from  the  glK  only  by  additive  constants, 

d?Xj  _  _  ^  dgu       ^  r  dgjj         dxi_/dg^  _  dg4i\  , 
dt*  2     dXi          L   dt  dt    \  dxi         dXi  / 


(23) 
dx3  dXi  /-I 

These  are  Newton's  equations  of  motion.  The  first  terms 
on  the  right  hand  represent  the  rectangular  components  of 
an  acceleration  which  is  the  gradient  of  a  newtonian  potential 


or,  vice  versa,  —£44  plays  the  role  (apart  from  an  additive 

2 
constant)  of  the  potential  multiplied  by  -  • 

c2 

The  second  terms  look  less  familiar.  But  their  meaning 
can  be  made  clear  at  once.  They  represent  at  any  rate  a 
certain  acceleration  field  which  need  by  no  means  be  negligible 
in  comparison  with  the  newtonian  one.  The  contributions  of 
this  field  to  the  components  of  acceleration  are 


F  dg*i   ,    dxz  /  dg4i        dg42  \  _  dx3   /  6g43  _  ^IfiY"!  etc 
L   dt  dt\dx2         dxi/         dt    \dxi         dxs/-\' 

or   in   ordinary   vector   language,  with   r=(#i,  #2,  ^3)   and 


4        TTT 
=  c  —^-    —  cv  —  curl  g4. 

dtz  dt  dt 


EQUATIONS  OF  MOTION  37 

This  is  manifestly  the  acceleration  due  to  a  velocity  field  eg* 
impressed  upon  the  system  of  reference.  If  this  velocity 
field  is  homogeneous  and  constant  in  time,  its  contribution  to 
acceleration  is,  of  course,  zero;  but  if  it  is  heterogeneous  and 
variable,  it  contributes  to  the  acceleration  of  a  free  particle 
through  its  time  rate  of  variation  and  through  the  vorticosity 
of  its  distribution.  The  simplest  case  occurs  when  g4  is  a 
linear  function  of  the  coordinates  alone,  say 

CO  CO 

g*l=-XZ,  g42=     --  Xi,  g43=0, 

C  C 

where  co  is  a  constant.  Then  c  curl  g4  is  a  (three-)  vector  of 
size  2co  directed  along  the  #3  —  axis  and  the  last  equation  gives 

2  2       _  __         dxi         dzxs   = 

° 


dt2  dt    '        dt2  dt  dt2 

which  is  the  Coriolis  acceleration  corresponding  to  a  uniform 
rotation  of  the  system  with  angular  velocity  co  round  the 

#3  —  axis  (vectorially,  with  the  angular  velocity  —  •  .  curlg4). 

2 

The  reader  will,  perhaps,  miss  the  centrifugal  acceleration 
coV,  Coriolis'  faithful  companion.  But  this  (having  a  scalar 
potential)  is  inseparable  from  g44.  It  is  included  in  g44  through 

the  term  --  ,  already  familar  to  us  from  a  previous  example. 
c2 

The  gu  just  given  will  be  found  by  noticing  that  in  (S),  p.  30, 
r"d6dx4=(xidx2  —  X2dxi)dx4.  This  settles  the  question. 

In  the  more  general  case  the  spin  \c.  curl  g4  will  not  be 
constant  but  will  vary  from  point  to  point  giving  rise  to  a  more 
complicated  acceleration  field.* 

The  approximate  equations  of  motion  (23)  can  now  be 
written  compactly,  in  three-dimensional  vector  language, 


_    =_v  +  c  \     -^-   -V— curlg4      .     (23a) 

dt2  2  L   dt  dt  J 

*I  propose  to  call  so  all  fields  corresponding  to  any  dsz,  and  to  reserve 
the  name  of  gravitational  fields  for  those  only  which  are  'permanent'  or 
cannot  be  transformed  away  holonomously. 


38  C  .    RELATIVITY  AND  GRAVITATION 

This  equation  brings  at  once  into  evidence  the  parts  played 
by  g44  and  by  the  three  £4;  condensed  in  g4-  Both  roles  may 
be  equally  conspicuous,  and  it  would  certainly  be  unjust  to 
say,  with  Einstein,  that  it  is  only  g44  which  survives  in  this 
first  approximation. 

Einstein  (loc.  cit.,  p.  817),  in  deriving  the  approximate  newtonian 
equations  from  the  rigorous  ones,  no  doubt,  through  a  too  hasty  computa- 
tion of  the  Christoffel  symbols,  dropped  altogether  the  second  terms  of 
our  equations  (23).  And  his  'slip'  crept  into  the  writings  of  de  Sitter, 
Weyl  and  others.  Einstein  exclaims  even  (ibid.)  in  genuine  surprise: 

dZXi  &         <3g44  "~| 

-  =  --     —  —    I  is  that  only 


dZXi  &         <3g44  "~| 

-  =  --     —  —    I  is 
dfi  2      oxi 


the  component  #44  of  the  fundamental  tensor  determines  by  itself,  in  a 
first  approximation,  the  motion  of  a  material  particle'. 

We  shall  return  to  these  approximate  equations  of  motion 
later  on,  after  having  set  up  Einstein's  gravitational  field- 
equations. 


UNIVERSITY  OF  CALIFORNIA 

DEPARTMENT  OF  CIVIL  ENGINEERING 

BERKELEY,  CALIFORNIA 


CHAPTER  III. 

Elements  of  Tensor  Algebra  and  Analysis. 


12.  In  order  to  be  able  to  construct  generally  covariant 
laws  or  equations,  such  as  Einstein's  field-equations  which 
will  complete  the  fundamental  part  of  his  theory,  some 
elementary  notions  of  the  Tensor  Calculus  are  required. 
These  I  shall  now  proceed  to  give,  without  stopping  to  sketch 
the  history  of  the  origin  and  the  growth  of  this  powerful 
method  of  multidimensional  analysis,  which  the  reader  will 
find  in  the  preface  to  Ricci  and  Levi-Civita's  paper  on  the 
Absolute  Differential  Calculus,*  as  the  said  branch  of  mathe- 
matics is  called  by  these  authors. 

The  relations  and  properties  which  are  now  to  occupy 
our  attention  hold  for  a  manifold  of  any  number  of  dimensions. 
But,  if  not  otherwise  stated,  we  shall  have  in  mind  our  four - 
dimensional  world  or  space-time. 

A  world-point  is  given  by  four  gaussian  coordinates  xt 
which,  in  general,  are  mere  numbers  or  labels.  As  such  they 
need  not,  as  in  the  most  familiar  treatment,  stand  for  such 
things  as  lengths  or  distances,  or  angles.  By  calling  them 
Mabels'  we  do  not  mean,  of  course,  that  tetrads  of  numbers 
are  being  haphazardly,  disorderly,  attached  to  various  events 

*G.  Ricci  and  T.  Levi-Civita,  Methodes  de  cakul  differentiel  absolu  et 
leurs  application,  Mathem.  Annalen,  vol.  54  (1900),  pp.  125-201.  A  con- 
densed account  of  this  paper  is  given  in  J.  E.  Wright's  Invariants  of  Quad- 
ratic Differential  Forms,  Cambridge  Tracts,  No.  9  (1908).  Perhaps  the 
easiest  presentation  of  all  that  is  required  for  relativistic  applications  is 
given  in  the  second  part  (B.)  of  Einstein's  own  paper,  loc.  cit,t  essentially 
reproduced  in  chap.  Ill  of  A.  S.  Eddington's  Report,  Phys.  Soc.  London, 
1918.  Th  subject  is  treated  on  original  and  very  attractive  lines  by  H. 
Weyl  in  Raum,  Zeit,  Materie  (Springer,  Berlin),  3rd  ed.,  1920.  For  geo- 
metrical applications  the  first  volume  of  L.  Bianchi's  Leziom  di  Geometria 
Differenziale  (Spoerri,  Pisa),  2  ed.,  1902,  can  be  most  warmly  recommended. 

39 


40  RELATIVITY  AND  GRAVITATION 

(world-points),  but  we  assume  that  #i  =  7,  say,  is  a  label 
attached  to  a  whole  connected  three-dimensional  continuum 
of  world-points,  and  similarly  for  all  other  (real)  numerical 
values  of  x\.  Likewise  for  the  remaining  coordinates,  so  that 
every  world-point  appears  as  the  intersection  of,  or  element 
common  to,  some  four  hypersurfaces  of  three  dimensions. 
Manifestly,  the  use  of  such  coordinates  does  not  presuppose 
any  idea  of  measurement.  Again,  in  this  abstract  treatment 
of  tensors  as  certain  entities  in  the  manifold,  the  question 
whether  any  one  of  the  coordinates  or  its  differential  is  space- 
like  or  time-like,  is  of  no  interest.  It  becomes  relevant  only 
when  we  come  to  apply  these  concepts  to  physical  problems. 
13.  Such  being  the  nature  of  the  x,,  pass  from  these  to 
any  other  coordinates  xj,  through  any  holonomous  transforma- 
tion whatever,  satisfying  only  the  conditions  of  continuity,  etc., 
as  stated  in  chapter  I.  Then,  as  in  (8a),  the  differentials  dxt, 
i.e.,  the  coordinates  of  a  world-point  Q,  a  neighbour  of 
P(xt),  with  P  as  origin,  are  transformed  into 

,    ,       dxt'  ,          dx{  ,          dx/  , 

dxL'  =  — -  dxK  =  — -  dxi  -\ dx2  +  .  .  .   . 

dxK  dxi  dxz 

That  is  to  say,  the  coordinates  of  Q  with  P  as  origin,  are  given, 
in  the  new  system,  by  these  linear  homogeneous  transforma- 
tions of  the  old  relative  coordinates  of  the  pair  of  points, 
with  coefficients,  dxt'/dxK,  which  are  some  given  functions  of 
the  position  of  P.  Such  an  ordered  point-pair,  PQ,  or  the 
corresponding  array  of  the  dxt,  is  called  a  vector,  in  our  case 
a  four-vector  or  world-vector.  From  a  more  general  standpoint 
to  be  explained  presently  its  name  is:  a  contravariant  tensor 
of  rank  one. 

Now,  as  in  special  relativity  every  tetrad  which  is  trans- 
formed as  the  cartesian  x,  y,  z  and  ct  (i.e.,  by  the  very  special, 
linear,  Lorentz  transformation),  so  here  the  tetrad  of  infini- 
tesimals dx,  is  made  the  prototype  of  all  (contravariant) 
vectors.  In  other  words,  every  tetrad  of  magnitudes  A1 
which  are  transformed  by  the  same  rule  as  the  dx,,  i.e., 

Afl=d-^AK,  (24) 

dxK 


COVARIANT  VECTORS  41 

is  called  a  contravariant  vector  or  a  contravariant  tensor  of  rank 
one,  and  A1,  A2\  etc.,  are  called  its  components.  (The  upper 
position  of  the  suffixes  was  proposed  by  Ricci  and  Levi-Civita 
and  accepted  by  all  authors.  To  be  consequent  one  would 
have  to  write  also  dxl,  as  in  fact  is  done  by  Weyl.  But,  for 
the  sake  of  typographical  convenience,  an  exception  is  being 
made  for  this  prototype  of  all  contravariant  vectors.)  It  is 
scarcely  necessary  to  say  that,  unlike  the  Cartesians  in  special 
relativity,  the  coordinates  xt  themselves  do  not  form  a  vector ; 
only  their  differentials  do.  In  short,  there  are,  in  general,  no 
finite  position-vectors,  but  only  differential  ones.  This,  how- 
ever, does  not  exclude  the  possibility  of  other  finite  vectors  A1. 

It  is  of  particular  importance  to  notice  the  linearity  and 
homogeneity  of  the  transformation  formula  (24)  which  will 
reappear  in  the  case  of  all  other  tensors.  The  all-important 
consequence  of  this  property  is  that  if  all  components  of  a 
vector  vanish  in  one  system,  they  will  vanish  also  in  all  other 
systems  of  coordinates.  More  briefly,  if  a  vector  AK  vanishes 
in  one  system  it  will  vanish  also  in  any  other  system.  Thus 
A*  =  Q  will  be  a  generally  'covariant'  or,  technically,  contra- 
variant  law.  This,  of  course,  does  not  prejudice  the  question 
whether  Nature  is  going  to  obey  it. 

Manifestly,  if  AK  and  BK  are  two  contravariant  vectors,  so 
also  are  AK+BK  and  AK-BK. 

As  dxt  served  as  the  standard  of  contravariant  vectors,  so 
do  the  operators  (differentiators) 

D-    d 
£ 

serve  as  a  prototype  of  another  kind  of  vectors.    We  have, 
evidently, 

D:=  ^DKt 

dxj 

and  every  tetrad  of  magnitudes  BL  which  are  transformed 
according  to  this  rule, 

BS-B.,  (25) 


42  RELATIVITY  AND  GRAVITATION 

is  called  a  covariant  vector  or  tensor  of  rank  one.  In  com- 
parison with  (24),  notice  that  the  suffix  of  B'  coincides  with 
the  lower  (instead  of  upper)  suffix  in  the  coefficients.  Although 
the  prototype  of  these  vectors  consists  of  differentiators,  the 
components  JB,  of  a  covariant  vector  need  not  be  operators,  but 
may  be  magnitudes  in  the  ordinary  sense  of  the  word.  As 
in  the  previous  case,  BK  =  Q  is  a  generally  covariant  equation 
or  rather  set  of  equations.  And  if  BK  and  CK  be  two  covariant 
vectors,  so  also  are  BK±CK.  Needless  to  say  that  AK-\-BK  is 
neither  a  covariant  nor  a  contravariant  vector.  In  fact,  it  has 
no  meaning  if  the  system  is  not  specified. 

14.  But,  while  the  sum  of  a  covariant  and  a  contravariant 
vector  is  from  the  present  point  of  view  of  no  interest,  the 
combination  of  their  components 


which  is  called  the  inner  or  scalar  product,  has  a  very  remark- 
able property.  It  is  invariant  with  respect  to  any  transfor- 
mations of  the  coordinates.  In  fact,  by  (24)  and  (25), 


dx!  d 

But  the  Xi,  #2,  etc.,  being  mutually  independent,  the  bracketed 
expression  (to  be  summed  over  all  i)  vanishes  for  all  K=F\  and 
equajs  1  for  K  =  X.  Whence, 

A'lBt'  =  AKBK  =  AlBt,  (26) 

which  was  to  be  proved. 

Any  invariant,  S=  Sf,  is  also  called  a  scalar  or  a  tensor  of 
rank  zero,  since,  in  a  manifold  of  n  dimensions,  it  has  n°  com- 
ponents, i.e.  but  one  component.  Similarly,  a  vector  or 
tensor  of  rank  one,  has  nl  =  n,  in  our  case  four,  components. 
The  question  whether  a  scalar  is  a  contravariant  or  a  covariant 
tensor  is  idle.  For  it  transforms  into  itself. 

Vice  versa,  it  can  easily  be  proved  that  if  BK  be  four 
(generally,  n)  magnitudes  such  that  A*  BK  is  invariant  for 
any  contravariant  A",  then  BK  is  a  covariant  vector.  And 


SECOND  RAN^  TENSORS  43 

the  same  thing  is  true  if  'covariant'  and  '  contra  variant ' 
be  exchanged  with  one  another. 

The  product  of  a  vector  by  a  scalar  is,  obviously,  again  a 
vector  of  the  same  kind,  and  any  number  of  vectors  of  the 
same  kind  multiplied  by  scalars  and  added  together  give 
again  a  vector  of  the  same  kind.  Finally,  notice  that  A KBK 
and  AKB"  are  not  invariant,  and  thus  are  no  tensors  at  all. 

15.  As  we  just  saw,  the  inner  multiplication  of  a  covariant 
and  a  contravariant  vector  degrades  the  rank  of  both  factors 
giving  a  tensor  of  rank  zero,  a  single  component.  Consider, 
on  the  other  hand,  what  is  known  as  the  outer  product  of  two 
vectors,  of  the  same  or  of  opposite  kinds,  i.e.,  AtBKt  or  A'B", 
or  AtBK.  The  suffixes  being  here  different,  no  summation  is 
understood,  so  that  each  of  these  symbols  stands  for  42=16 
(generally  n^  components.  Let  us  take  AtBK  first,  which  is  a 
short  symbol  for  the  array 


of  sixteen  magnitudes.  Denote  them  by  MIK  respectively. 
Their  law  of  transformation  is,  by  (25), 

;      '       M-^M«-\'  (27) 

Every  array  of  n*  magnitudes  N^  (whether  obtained  by  the 
outer  multiplication  of  two  covariant  vectors  or  in  any  other 
way)  which  is  transformed  by  the  rule  (27)  is  called  a  covariant 
tensor  of  rank  two.  It  manifestly  has  again  the  property  of 
vanishing  in  all  systems,  if  it  vanishes  in  one  of  them.  In  a 
four-manifold  NIK  consists  of  16  components. 

In  general  N«  4=  NKt.  If,  in  particular,  NtK=  NKl  the  tensor 
is  called  symmetrical. 

An  example  of  such  a  tensor  we  had  in  giKJ  called  the 
fundamental  tensor;  cf.  formula  (9).  Notice,  however,  that 
the  tensor  property  of  glK  followed  from  the  invariance  of  ds* 
which  fixed  the  metrical  properties  of  the  world,  whereas  all 
our  present  considerations  are  entirely  independent  of  the 

—4 


44  RELATIVITY  AND  GRAVITATION 

metrics  of  the  manifold,  and  it  is  preferable  to  abstain  from 
using  them  at  this  stage.  Such  properties  as  are  impressed 
upon  the  general  tensors  by  the  metrics  of  the  world  will  be 
treated  in  later  sections. 

In  the  meantime  let  us  continue  the  non-metrical  theory  of 
tensors. 

The  symmetrical  tensor  NLK  consists  in  general  of  Jw(w  +  l), 
and  for  w  =  4,  of  ten  different  components.  It  can  be  easily 
proved,  by  (27),  that  its  symmetry  is  an  invariant  property, 
i.e.,  that  if  Nllc=  NKt  in  one  system,  we  have  also  N'uc=NfKt  in 
any  other  system.  A  covariant  symmetrical  tensor  of  rank 
two  can  be  constructed  at  once  from  a  covariant  vector,  to 

wit  by  forming  its  outer  self-product,  Aflv  =  A(tAl>  =  AvAtt=: 

A 
*»tp« 

If  NIK=  —NKi,  for  all  i,  K,  we  have  an  antisymmetrical  (or 
skew)  tensor.  Since  NKK=  —  NKK  means  NKK  =  Q,  a  whole 
diagonal  of  components  vanish,  and  thus  only  %n(n+l)  —  n  = 
\n(n—l)  non-vanishing  and  independent  components  are 
left,  the  surviving  ones  being  oppositely  equal  in  pairs. 
Thus  an  antisymmetric  tensor  in  a  four-  world  consists  of 
six  independent  components,  and  is  therefore  called  a  six- 
vector,  in  the  present  case  a  covariant  six-  vector.  With  such 
six-vectors  the  reader  is  already  acquainted  from  the  special 
relativistic  treatment  of  the  electromagnetic  field.  We  shall 
see  them  at  work  in  a  similar  duty  in  general  relativity 
later  on. 

As  the  symmetry  so  also  the  antisymmetry  is  an  invariant 
property,  i.e.,  NiK  =  —  NKl  is  transformed  into  N'LK=  —  N'Kt. 

Any  tensor  NiK  can  be  split  at  once  into  a  symmetrical 
and  an  antisymmetrical  one.  For  we  have  identically 


and  the  first  term  represents  a  symmetrical,  the  second  an 
antisymmetrical  tensor. 

Similarly  to   (27),  and  starting  from  the  special  tensor 
A1B",  any  array  of  n2  magnitudes  which  are  transformed  by 

the  rule  7Vm  =  ^L    dx-±-  Nafi  (27  a) 

dx        dx 


MIXED  TENSORS  45 

v    . 

is  called  a  contravariant  tensor  of  rank  two.  If  NUC=NK\  it  is 
a  symmetrical,  and  if  Nuc  =  —TV"1,  an  antisymmetrical  tensor. 
(A  tensor  N™  need  not  be  the  product  of  two  contravariant 
vectors.) 

Lastly  (starting  from  ALBK),  any  array  of  n2  magnitudes 
N*  which  are  transformed  by  the  mixed  rule 


- 

dxj     dxa 


(276) 


is  called  a  mixed  tensor  of  rank  two,  covariant  with  respect  to 
its  lower  suffix  t,  and  contravariant  with  respect  to  its  upper 
suffix  or  index  K*  Special  cases  of  symmetry  and  anti- 
symmetry as  before.  A  new  feature,  however,  offered  by 
the  mixed  tensor  is  this.  With  any  N\  make  I  =  K,  getting 
NKK  and,  by  the  usual  convention,  sum  over  all  K.  In  other 
words  add  up  all  the  components  of  the  chief  diagonal  (slanting 
down  from  left  to  right)  of  the  mixed  tensor.  The  result  will 
be  a  single  magnitude.  Now,  the  important  thing  is  that 
this  magnitude  is  a  general  invariant.  In  fact,  by  (276), 

AT/*:  /       UXft        vX»         1  ATd    . 

N  *  =  I    f     jMff  •' 

\dx'K    dxa/ 

but  (as  mentioned  before)  the  bracketed  expression  is  zero 
for  all  a?^j8  and  one  for  a  =  /3.  Thus 

N'KK=N:=NKKJ 

which  proves  the  proposition. 

Thus,  equalling  the  upper  and  the  lower  index  and 
summing  over  it  degrades  the  mixed  tensor  by  two  ranks 
giving,  in  the  present  case,  a  tensor  of  rank  zero  or  an 
invariant  (scalar).  In  other  words, 

NKK=N 


*It  seems  inappropriate  to  call  'suffix'  (from  sub,  under)  an  upper  mark 
or  sign.  I  propose  .therefore,  to  call  such  signs  by  the  more  general  name 
index.  Since  all  English  writing  authors  accepted  the  '  three-index  symbols ' 
and  the  'four-index  symbols'  (of  Christoffel  and  Riemann),  they  will  per- 
haps not  object  to  calling  t,  K  indices. 


46  RELATIVITY  AND  GRAVITATION 

is  an  invariant  of  the  tensor  NKt.  We  shall  see  presently  that 
this  procedure  of  equalling  an  upper  to  a  lower  index,  called 
contraction  (German  '  Verjungung')  can  be  applied,  with  equal 
success,  to  a  mixed  tensor  of  any  rank  whatever.  Notice, 
however,  that  this  process  is  not  applicable  in  the  case  of 
(purely)  covariant  or  contravariant  tensors.  Thus,  for 
instance,  MKK  =  Mi\~\-Mm-{-  ...  is  not  invariant,  as  a  glance 
on  (27)  will  suffice  to  show.  In  short,  the  diagonal  sum  of 
MIK  has  no  intrinsic  meaning.  Similarly,  in  the  case  of  a 
four-  vector,  say,  AI+  .  .  .  -\-A4  is  not  an  invariant. 

16.  The  next  step,  leading  to  tensors  of  rank  three,  and 
so  on,  is  obvious.  Generally,  any  system  of  nr  (in  our  world,  4r) 
magnitudes  N£\\',  with  r\  lower  and  r%  upper  indices,  which 
are  transformed  by  the  rule 


dx.'     dxK'     dxa      dx 


is  called  a  mixed  tensor  of  rank  r  =  n+f2,  covariant  with  respect 
to  its  fi  lower,  and  contravariant  with  respect  to  its  r2  upper 
indices.  If  all  the  components  of  such  a  tensor  vanish  in  one 
system  they  will  also  vanish  in  any  other  system  of  coordinates. 
Any  tensor,  therefore,  can  be  used  for  writing  down  generally 
covariant  laws.*  In  particular,  if  ri  =  0,  the  tensor  (28)  is  con- 
travariant, of  rank  r2;  and  if  r2  =  Q,  covariant  of  rank  r\.  The 
sum  of  any  number  of  tensors  of  the  same  rank  and  kind, 
each  multiplied  by  any  scalar,  is  again  a  tensor  of  the  same 
rank  and  kind,  the  numbers  n,  r2  retaining  their  significance. 

17.  Contraction.  This  process,  already  illustrated  on  the 
simplest  example,  can  now  be  generally  explained. 

Let  a  be  any  upper  and  i  any  lower  indexf  of  a  mixed 
tensor  of  any  rank  r  whatever.  Put  a  =  t  and  sum  over  a. 
Then  the  result  will  be  a  tensor  of  rank  r  —  2,  with  r\—  1  covari- 
ant and  rz—  1  contravariant  indices. 


*In  the  less  technical  sense  of  the  word. 

fThe  place  of  a  among  the  upper,  and  of  t  among  the  lower  indices  is 
irrelevant. 


CONTRACTION  OF  TENSORS  47 

The  proof  follows  at  once  from  (28).    For  the  process  gives 
us  in  the  coefficients  of  transformation  a  term 

dxa     dxt' 
dxt'      dXi 

which  vanishes  for  all  a  7^1  and  equals  one  for  a  =  i,  thus 
reducing  (28)  to 


. 

dxK'     dxk 

which  proves  the  statement. 

This  process  of  contraction  can  obviously  be  applied  again 
and  again,  degrading  the  tensor  each  time  by  two  ranks  until 
there  will  be  no  upper  or  no  lower  indices  left.  In  fine,  the 
mixed  tensor  can  be  degraded  until  it  becomes  purely  covariant 
or  purely  contra  variant  or  (if  ri  =  rs)  until  it  is  reduced  to  a 
scalar  or  invariant. 

Thus,  for  example,  the  tensor  A^  of  rank  five  gives  rise  to 


which  is  denoted  by  A^  ,  and  this  tensor  of  rank  three  gives 
rise  to 

A'A  =  A-, 

which  is  a  (covariant)  tensor  of  rank  one  or  a  vector. 

Again  (as  an  example  of  r\  =  rz)  ,  the  tensor  A  jf  of  rank  four 
gives  by  contraction  A%  ,  and  this  tensor  of  rank  two  gives 


a  scalar.  We  may  as  well  write  at  once  A™  =  A,  the  meaning 
and  the  value  of  A  being  the  same  as  before.  This  final 
invariant  may  be  considered  as  a  property  of  the  original 
tensor  A*  . 

In  general  every  such  half-and-half  tensor  (r\  =  r^)  will  have 
the  final  scalar  (A)  as  its  intrinsic*  invariant.  And,  as  far  as 
I  can  see,  this  is  its  only  intrinsic  invariant. 

**.«.  an  invariant  of  its  own,  independent  of  any  extraneous  form  such 
as  ds*  (or  any  auxiliary  tensor,  such  as  glK  )  determining  the  metrics  of  the 
manifold. 


48  RELATIVITY  AND  GRAVITATION 

On  the  other  hand  a  purely  covariant  or  contravariant 
tensor  or  an  unequally  mixed  one  (fi^fz)  cannot  be  contracted 
to  an  invariant.  It  seems  that  it  has  no  intrinsic  invariant 
at  all,  that  is  to  say,  that  there  are  no  processes  which 
would  lead  to  an  invariant  combination  of  the  components 
of  the  original  tensor  itself  (without  using  other  tensors). 

18.  The  inner  multiplication,  already  mentioned  in  con- 
nection with  vectors,  can  now  be  considered  as  an  outer 
multiplication  followed  by  a  contraction. 

Consider  two  tensors,  generally  mixed,  one  of  rank  r  =  fi+ 
rz,  the  other  of  rank  5  =  Si+s2.  Combine  (by  ordinary  multi- 
plication) each  of  the  nr  components  of  the  former  with  each 
of  the  ns  components  of  the  latter.  The  nr+s  magnitudes  thus 
obtained  will  be  the  components  of  a  tensor  of  rank  r+s 
with  n+5i  covariant  and  TI+SZ  contravariant  indices.  That 
the  entity  thus  arising  is  a  tensor  follows  at  once  from  (28). 

Thus  the  outer  product  of  two  vectors  is  a  tensor  of  rank 
two,  A.BK  =  MIK,  A1BK  =  MKL  .  Similarly  AafiBtK  is  a  covariant 
tensor  of  rank  four,  Ma(3lK,  and  AafiB"  =  N"py  is  a  mixed 
tensor  of  rank  five,  and  so  on. 

The  outer  multiplication  combined  with  contraction 
(when  there  are  indices  to  contract)  gives  the  inner  product. 
Thus  the  inner  product  of  At  and  BK  is 

AKBK=MKK  =  M, 

an  invariant.*  The  inner  product  of  A"  and  Bap  is  their  outer 
product  Mrf  degraded  by  contraction,  i.e.,  M^  =  Ma,  a  covari- 
ant vector.  The  inner  product  of  A^  and  BIK  is  their  outer 
product  Aa0Buc=M%i  degraded  (to  the  utmost)  by  two 
contractions, 

M?K  =  M, 

i.e.,  a  scalar  or  invariant.  Vice  versa,  if  Aap  be  any  array  of 
ri*  magnitudes  such  that  A^B™  is  an  invariant  for  any  con- 
travariant B™,  then  A^  is  a  covariant  tensor  of  rank  two. 
This  criterion  of  tensor  character,  already  mentioned  in  con- 
nection with  AtBK,  can  be  easily  proved  by  writing  down  the 

There  is  no  inner  product  of  At  ,  BK  .  • 


TENSOR  DIFFERENTIATION  49 

transformation  formula  of  the  given  factor  (tensor).  And  it 
can  be  extended  to  any  rank  and  kind,  no  matter  whether 
the  inner  product  is  a  scalar  or  a  tensor  of  any  rank  higher 
than  zero. 

As  we  already  know,  the  differential  operators  Dt  =  d/dxt 
have  the  character  of  the  components  of  a  covariant  tensor 
of  rank  one.  Therefore,  the  'product'  of  this  tensor  into  a 
scalar  or  scalar-field/=/(xi,  x2  .  -  .)»  that  is  to  say,  the  result 
of  operating  with  Di  upon/,  will  again  be  a  covariant  tensor 
of  rank  one  or  a  covariant  vector, 


(29) 


But  we  cannot  go  further  than  that.  That  is  to  say,  an  iterated 
application  of  the  operation  DK  does  not  give  a  tensor.  Thus 
d2f/dxtdxK  is  not  a  tensor.  Nor  do,  in  the  more  general  case 
of  any  vector  Bt,  the  n2  derivatives  DKBl  =  dBJdxK  constitute  a 
tensor.  The  different  behaviour  of  DKBL  and  of  products  of 
magnitude-tensors  lies  herein  that  the  operational  tensor  DK 
acts  also  on  the  coefficients  dxjdxj  of  the  transformation 
formula  of  Bt.  In  fact,  we  have 


a*»' 

and*  this  is  not  the  same  thing  as  — - — —  DaBa-  The  same 

dxK'  dx/ 

remark  applies,  a  fortiori,  to  higher  derivatives  of  scalars  and 
of  tensors  of  any  rank. 

In  fine,  the  only  tensor  derivable  by  simple  differentiation, 
unaided  by  other  auxiliaries  (cf.  infra),  is  the  covariant  vector 
(29)  yielded  by  a  scalar.  The  vector  or  vector-field  df/dxt  is 
called  the  gradient  of  /.  In  the  case  of  space-time  it  consists 
of  four  components. 


*Unless  the  coordinate  transformations  are    linear  as  in  the  special 
relativity  theory. 


50  RELATIVITY  AND  GRAVITATION 

19.  Tensor  properties  in  a  metrical  manifold.  Having 
sufficiently  acquainted  ourselves  with  the  properties  of  tensors 
in  themselves,  let  us  now  consider  them  in  relation  to  the 
fundamental  quadratic  form  ds2  =  glK  dxt  dxK  which  converts 
the  hitherto  amorphous  world  into  a  metrical  or  riemannian* 
manifold. 

It  is  of  the  utmost  importance  to  grasp  well  this  distinction 
between  a  riemannian  and  a  non-metrical  manifold  and  to 
understand  the  true  role  of  ds2  in  converting  the  latter  into  the 
former. 

Let  us  place  ourselves  yet  for  a  while  upon  the  non-metrical 
standpoint.  Of  all  the  tensors  described  in  the  preceding 
sections  let  us  confine  our  attention  upon  the  prototype  of  all 
(contravariant)  vectors,  the  infinitesimal  position-vector  dxt. 
Any  such  vector  represents  ultimately  but  an  ordered  pair  of 
points,  0(X)  the  origin,  and  A(xi+dxi)  the  end-point  of  the 
vector.  Imagine  a  whole  bundle  of  such  infinitesimal  vectors 
OA,  OB,  OC,  etc.,  all  emerging  from  the  same  world-point 
O  as  origin.  Now,  from  the  non-metrical  point  of  view,  all 
these  vectors  have  (apart  from  their  origin)  nothing  in  common 
with  one  another.  That  is  to  say,  if  two  of  them,  say  OA 
and  OB,  are  at  all  distinct  from  one  another,  and  if  their 
components  dxL  do  not  happen  to  be  proportional  to  one 
another  (in  which  case  we  can  say  that  the  vectors  have  a 
common  'direction'),  there  is  in  either  of  them  nothing,  no 
property,  with  respect  to  which  they  could  be  compared.  They 
are,  as  it  were,  perfect  strangers  to  one  another.  Similarly,  if 
we  call  'angle'  a  vector-pair  a  =  OA,  OB,  there  is  nothing  to 
base  upon  a  comparison  of  two  non-overlapping  covertical 
angles  a  and  /3  =  OC,  OD.  In  short,  neither  vectors  nor  angles 
(or  other  derived  entities)  have  'sizes'.  There  is,  in  fact,  in 
the  manifold  itself  nothing  which  could  fix  the  mere  meaning 
of  suqh  a  concept.  Of  two  vectors  OA ,  OB  nothing  more  can 

The  name '  riemannian '  manifold  or  w-space  is  being  often  used  in  this 
connection  in  view  of  the  historical  fact  that  Riemann  was  the  first  to  base 
the  general  geometry  of  an  w-space  upon  its  line-element  given  by  such  a 
differential  form,  although  Gauss  was  his  great  predecessor  in  the  case  of 
surface  theory. 


THE  LINE-ELEMENT  51 

be  said  than  that  they  are  either  identical  (or  co-directional, 
collinear)  with  or  distinct  from  one  another.  The  origin 
being  the  same,*  the  points  A,  B  are  either  identical  or  dis- 
tinct, and  no  other  significant  statement  can  be  made  about 
their  relation. 

But  while  there  is  nothing  in  the  manifold  itself  to  base  a 
comparison  of  distinct  infinitesimal  vectors  upon,  we  are  at 
liberty  to  provide  for  it  at  our  will  if  we  so  desire.  This  is 
done  by  introducing  a  standard  or  fundamental  entity  such 
as  the  quadratic  form  called  the  line-element.  In  other 
words,  we  surround  the  world-point  0(xt)  by  a  hypersurface, 
a  three-dimensional  (generally  an  n  —  I  dimensional)  quadric 
and  declare  all  vectors  emerging  from  O  and  ending  in  any 
point  P  (xt-\-dxt)  of  this  surface  to  be  equal  in  size  or  in 
absolute  value,  or  in  'length',  the  usual  name  in  the  case  of 
our  three-space.  It  is  precisely  this  metrical  surfacef  which 
is  expressed  by 

gudx^Xt  =  ds2  =  const., 

the  numerical  value  of  ds  being  the  'size'  common  to  all  these 
infinitesimal  vectors  or  point-pairs. J  The  part  played  by 
this  quadratic  form  is  essentially  the  same  as  that  of  Cayley's 
'absolute'  or  standard  quadric  (a  real  quadric  leading  to 
lobatchevskyan  or  hyperbolic,  an  imaginary  quadric  leading 
to  elliptic,  and  the  intermediate  degenerate  quadric  leading 
to  euclidean  geometry),  the  only  important  difference  being 
that  Riemann's  treatment  is  much  more  general.  It  covers 

*We  have  limited  the  discussion  to  coinitial  vectors  solely  for  the  sake  of 
simplicity.  All  our  remarks  apply  a  fortiori  to  distant,  non-coinitial 
bundles  of  vectors. 

fThe  German  geometers  call  it  Eichflache. 

Jin  Riemann's  own  treatment  this  r61e  of  the  fundamental  form  im- 
pressed upon  the  manifold  extends  into  distance,  over  all  the  manifold. 
That  is  to  say,  if  O'(y^  be  any  other  point  and  if  a  quadric  glKdydyK  =  const, 
be  drawn  around  it  with  the  same  value  of  the  constant  as  before,  all  the 
vectors  of  the  bundle  O'  terminating  upon  this  quadric  are  again  said  to 
have  the  same  size  as  those  of  the  bundle  0.  In  this  respect  a  somewhat 
more  general  standpoint  was  recently  proposed  by  Weyl,  in  connection 
with  his  ideas  on  electromagnetism. 


52  RELATIVITY  AND  GRAVITATION 

all  metrical  spaces  (in  technical  language,  of  variable  and 
anisotropic  curvature),  whereas  Cayley's  device  gives  us 
only  a  space  of  constant  isotropic  curvature,  negative,  zero, 
or  positive.  This  fully  corresponds  to  his  starting  point, 
which  was  that  of  projective  geometry.  Yet,  and  this  is  of 
particular  interest  in  the  present  connection,  Cayley  recog- 
nized thoroughly  the  true  r61e  of  all  such  standard  entities. 
In  fact,  he  tells  us  plainly  that  geometrical  figures  have  no 
metrical  properties  in  themselves.  Their  metrical  properties 
such  as  those  of  the  foci  of  a  conic,  etc.,  arise  only  by  relating 
them  to  other  figures,  as  the  'absolute'  conic  in  the  plane,  or 
quadric  in  three-space. 

The  kind  of  metrics  thus  impressed  upon  a  continuous 
manifold  being  essentially  arbitrary,  the  utility  of  the  metrical 
manifold  thus  obtained  will,  of  course,  from  the  physicist's 
standpoint,  depend  upon  the  interpretation  which  is  given  to 
the  said  'size'  of  a  position-vector,  and  to  special  lines  of 
that  metrical  manifold,  such  as  the  geodesies,  in  terms  of 
measuring  rods,  clocks,  moving  particles  or  light  phenomena, 
and  so  on. 

But  without  dwelling  here  any  further  upon  such  questions 
of  a  concrete  representation  let  us  turn  to  consider  the  purely 
mathematical  consequences  of  the  introduction  of  g«  dxt  dxK 
as  a  fundamental  differential  form  fixing  the  metrics  of  the 
manifold. 

20.  As  in  Cayley's  case  the  geometrical  figures  in  relation 
to  his  'absolute',  so  here  the  tensors  acquire  some  new  pro- 
perties in  relation  to  the  fundamental  form  or  better,  to  its 
coefficients  glK .  In  fact,  what  determines  the  form  are  these 
coefficients,  and  we  may  look  upon  the  matter  in  the  following 
way. 

Instead  of  declaring  the  fundamental  quadratic  form  at  the 
outset  as  an  invariant,  let  us  better  say  that  the  symmetrical 
array  of  16  (generally  n2)  magnitudes  glK  is  being  introduced 
as  &  fundamental  tensor,  symmetrical,  of  rank  two  and  of  the 
covariant  kind,  as  defined  in  the  preceding  sections. 

Combined  with  this  fundamental  tensor  all  other  tensors 
of  the  previously  amorphous  manifold  will  acquire  some 


METRICAL  PROPERTIES  53 

new  properties.     These  and  only  these  will  now  be  their 
metrical  properties. 

To  begin  with  the  prototype  of  contravariant  vectors,  the 
infinitesimal  vector  dxL  has  had  thus  far  no  invariant  of  his 
own.  But  it  will  acquire  one  with  the  aid  of  the  fundamental 
tensor.  In  fact,  dxt  being  contravariant,  denote  it  for  the 
moment  by  X1.  Form  the  outer  product 


which  will  be  a  mixed  tensor  A^  .  Contract  it  with  respect 
to  t,  a,  getting  A^—A^.  Contract  this  again.  Then  the 
result  will  be  A"  =  A,  a  scalar  or  invariant.  Or  perform  both 
contractions  at  once,  and  write  dsz  for  A,  returning  to  the 
original  notation,  thus 

gu  dxt  dxK  =  dsz  =  invariant. 

In  short,  the  inner  product  of  the  tensor  dxa  dx$  into  the  funda- 
mental tensor  gtK  is  an  invariant.  There  is  no  objection  to 
calling  it  the  invariant  of  dxt  as  a  short  name  for  its  metrical 
or  associated  invariant.  Thus,  thanks  to  giK  ,  the  vector  dxt 
has  acquired  an  invariant.  And  it  can  now  be  compared 
through  it  with  other  vectors,  no  matter  what  their  com- 
ponents. The  value  of  ds2  may  be  called  the  norm,  and  the 


absolute  value  of  dbds2  the  size  of  the  vector  dx,..  Thus 
we  can  speak  of  two  vectors  dxt  and  dyt  being  equal  in  size, 
or  one  having  twice  the  size  of  the  other,  and  so  on.  In 
application  to  the  four-  world,  a  vector  dx,  of  no  size  will  be 
a  light  vector,  a  vector  of  negative  norm  a  space-like,  and 
one  of  positive  norm  a  time-like  vector. 

Similarly,  any  other  contravariant  vector  A1  will  have  the 
metrical  invariant 

glKA>AK=A\  say.*  (30) 


*Of  course,  even  in  the  amorphous  manifold  an  invariant  could  be  built 
up  from  A1  by  the  aid  of  any  covariant  tensor  NtK,  but  the  choice  of  NIK 
being  entirely  free,  such  an  invariant  would  not  have  a  fixed  value.  We 
fix  it  by  introducing  once  for  all  a  special  tensor  glK  to  serve  for  all  other 
tensors. 


54  RELATIVITY  AND  GRAVITATION 

In  much  the  same  way,  if  Bt  be  any  covariant  vector,  we 
shall  have  in 

g^B.B^B2  (30o) 

an  invariant,  the  norm  of  B.  . 

From  a  more  general  point  of  view  we  may  call  A2,  in  (30)  , 
the  tensor,  of  rank  zero,  metrically  associated  to  A1,  similarly, 
in  (30a),  B2toBt. 

Moreover,  we  can  easily  construct  associated  tensors  of  a 
rank  other  than  zero,  and  differing  also  in  kind  from  the 
original  tensor.  Thus,  to  dwell  still  upon  vectors, 

&.A--A.  (31) 

will  be  the  covariant  vector  metrically  associated  with  the 
contra  variant  vector  AK.  We  may  call  A,  shortly  the  conjugate 
of  A1.  Similarly,  starting  from  a  covariant  vector  A^  we 
shall  have  the  contravariant  vector 

gKtAK  =  Al  (31a) 

conjugate  to  At. 

Two  questions  naturally  suggest  themselves:  Will  the  con- 
jugate of  the  conjugate  be  the  original  vector?  Have  two 
conjugate  vectors  the  same  size  or  the  same  norm? 

In  order  to  answer  these  questions  as  well  as  for  the  sake 
of  what  will  follow,  let  us  first  note  a  simple  property  of  the 
tensors  glK  and  glK  .  By  definition,  chap.  II,  glli  is  the  minor 
of  the  determinant  g  =  |  gtK  \  ,  corresponding  to  its  t,  K-th  element, 
divided  by  g  itself.  But  g  is  equal  to  the  sum  of  the  products 
of  the  elements  of  its  first  column,  say,  into  the  corresponding 
minors,  i.e.,  g  =  gai  ggai,  whence  gaigai==  1.  Similarly  for  any 
other  column  (or  row).  Thus,  underlining  the  index  over 
which  an  expression  is  not  to  be  summed, 


This  is  valid  for  every  v.  Thus  gMJ,gM",  summed  over  both 
indices,  has  the  value  4  for  our  world,  and  n  for  an  w-fold. 
Again,  taking  two  different  columns  (or  rows)  of  g,  we  shall 
easily  prove  that 


CONJUGATE  TENSORS  55 

Both  properties  can  be  united  in  a  single  formula 

&.«"-«;-*!!,  (32) 

where  5«  is  the  conventional  symbol  for  1  or  0  according  as 
a  =  /3  or  a  5^/3.    This  symbol  is  itself  a  mixed  tensor. 

We  are  now  able  to  answer  our  two  questions.  First,  the 
conjugate  of  the  conjugate  of  the  vector  A  t  is,  by  the  definitions 
(31), 


i.e.,  the  original  vector.     Similarly  if  we  started  with  A". 
Thus,  the  conjugate  of  the  conjugate  is  the  original  vector. 

Second,  if  A  l  be  the  conjugate  of  A  t  we  have  for  the  norm 
of  the  former  vector,  by  (30)  and  (3  la), 


Thus  any  two  conjugate  vectors  have  equal  norms. 

The  norm  of  At  and  of  A'  can  also  be  written  AtAl,  for 
this  is  again  equal  to  gllf  A1  A".  Thus,  for  instance,  if  d^  be  the 
conjugate  of  the  contravariant  vector  dxL,  their  common 
norm  or  the  squared  line-element  can  be  written 

ds*  =  dxtd&.  (33) 

21.  In  much  the  same  way  we  can  treat  the  metrical 
properties  of  tensors  of  any  higher  rank.  To  explain  the 
method  it  will  be  enough  to  take  up  in  some  detail  the  second 
rank  tensor  At/c  .  Its  conjugate  or  supplement  (Erganzung) 
will  be  the  contravariant  tensor  defined  by 


A"  ,  or  also  g^g^A'A^  .  (34) 

The  tensor  g™  itself  is  easily  proved  to  be  the  supplement  of 
the  tensor  gu  . 

The  scalar  or  invariant  of  AIK  will  be 

fAu=Al-A.  (35) 

A  single  contraction  of  g"  Aaff  will  give 

g"  Am.  =  A',  , 
a  mixed  tensor  metrically  associated  with  the  covariant  A,,. 


56  RELATIVITY  AND  GRAVITATION 

The  supplement  of  the  supplement  (or  the  conjugate  of  the 
conjugate)  is  again  the  original  tensor,  for 


The  tensors  ALK  and  A"  have  the  same  scalar  A  ,  (35).  In  fact, 
the  scalar  of  A"  is 

a.  A"  -a.  g"  f'A^-P.  fA.t  =  f'At.~A. 

Since  gvApV  is  an  invariant,  ^  =  glK  £*  A^v  is  again  a  tensor; 
Einstein  calls  it  the  reduced  tensor  belonging  to  A^. 

Notice  that  neither  a  covariant  nor  a  contravariant  tensor  has  an 
invariant  independent  of  the  metrical  tensor;  only  a  mixed  tensor,  B"^ 
has  such  an  invariant,  to  wit  B  =  5\  This  is  a  privilege  of  mixed  tensors  of 
even  rank  with  ri=rz,  and  of  these  tensors  only. 

The  investigation  of  other  metrical  properties  of  tensors 
of  the  second  and  higher  ranks  may  be  left  to  the  reader. 
Exercises  of  such  a  kind  will  soon  make  him  familiar  with  this 
broad  and  powerful  algorithm. 

22.  Angle  and  volume.  Consider  any  two  coinitial  in- 
finitesimal vectors  dxt,  dyt.  These  are  contravariant  vectors. 
Therefore,  as  we  already  know,  the  inner  product 

gtK  dx,  dyK 

will  be  an  invariant.  It  will  remain  invariant  when  divided 
by  the  sizes  of  both  vectors.  By  an  obvious  generalisation  of 
the  familiar  cosine  formula  this  invariant  is  used  to  define 
the  angle  e  made  by  the  two  vectors,  thus 

cos  t  i  il^i^l  ,  (36) 

as  da 

where  ds2  =  glK  dxt  dxK,  da2  =  gLK  dyt  dyK,  The  two  vectors  are 
said  to  be  orthogonal  or  perpendicular  upon  one  another  if 

gtKdxtdyK  =  0. 

Generally,  the  angle  between  any  two  vectors  A1,  B\  whose 
norms  as  defined  by  (30)  are  A2  and  B2,  will  be  determined  by 


ANGLE  AND  VOLUME 


57 


COS  €   = 


AB 


(37) 


and  the  vectors  will  be  orthogonal  if  gu  ^4M'(  =  0.  Similarly 
for  covariant  vectors,  with  the  only  difference  that  gtK  is 
replaced  by  g1*  .  Let  Alt  Bt  be  the  conjugates  of  A1,  Bl;  then 


and  since  At,  Bt  have  the  same  norms  as  A1,  Bl,  we  see  that 
the  angle  between  the  conjugates  is  the  same  as  between  the 
original  vectors. 

The  integral  fdxi  dx2  .  .  .  dxM  extended  over  a  domain  of 
the  manifold  is,  by  a  well-known  theorem,  transformed  into 


i  dxz'  .  .  .  dxn',  where  J  is  the  Jacobian 


dxt 


,  as  in  (7). 


On  the  other  hand  the  determinant  g  of  the  fundamental 
tensor  (called  also  the  discriminant  of  the  fundamental 
quadratic  form)  is  transformed  into 

dxn        dxa 


a*.'    dx," 

the  last  step  being  based  on  the  multiplication  rule  of  deter- 
minants.    Thus 

g'  =  J*g.  (38) 


Consequently,  the  integral 


i  dx2  .  .  .  dxn 


(39) 


is  a  scalar  or  an  invariant  of  the  n  —  dimensional  domain  of 
integration. 

In  the  case  of  the  four-dimensional  world  the  determinant 
g  is  always  negative.*      Thus  the  invariant  expression 

*In  a  galilean  domain  and  in  Cartesians  g=  —1,  by  (16),  p.  6.  By  (38), 
therefore,  it  is  also  negative,  always  for  a  galilean  domain,  in  any  other 
system  of  coordinates  derived  from  the  Cartesians  by  a  holonomous  trans- 
formation. Now,  although  a  non-galilean  domain  cannot  be  made  galilean 
by  a  holonomous  transformation,  yet  we  know  that  in  all  practical  cases 
the  gLK  differ  but  very  little  from  the  galilean  coefficients.  Thus  g  will  also 
in  general  be  negative. 


UNIVERSITY  OF  CALIFORNIA 
DEPARTMENT  OF  CIVIL.  ENGINEERING 


58  RELATIVITY  AND  GRAVITATION 

dtt  =  V-g  dxidx2dxzdxi  (40) 

will  be  real.  This  is  taken  as  'the  local  measure'  of  the  size 
or  volume  of  an  infinitesimal  world-domain.  For  in  the  local 
(cartesian)  coordinates  ult  for  which  g=  —  1,  this  expression 
becomes  duidu2dMzdu^  =  cdtdxdydz.  The  latter  product  is 
called  by  Einstein  'the  natural'  volume-element.  Apart 
from  names,  the  important  thing  to  notice  is  the  general 
invariance  of  the  expression  (40)  as  such  or  when  integrated 
over  any  world-domain. 

Consider  any  sub-domain  of  the  world,  of  three,  two  or  one  dimension. 
This  can  be  represented  by  expressing  the  xt  as  functions  of  three,  two  or 
one  parameter  respectively.  The  differentials  dxt  will  be  homogeneous 
linear  functions  of  the  differentials  dpa  of  these  independent  parameters. 
Thus  the  line-element  within  the  sub-domain  will  be  of  the  form 

ds*  =  hapdpa  dpp  ,  hap  =  hpa, 

and  the  sub-domain,  therefore,  will  again  be  a  metrical  manifold  (a  three- 
space,  surface  or  line)  in  Riemann's  sense  of  the  word,  and  if  /*  =  | 

dQ^V h  dpidp, ..  . 


will  (apart  perhaps  from  a  factor  V  —  1)  again  be  an  invariant  measure  of 
an  element  (volume,  area,  length)  of  the  sub-domain. 

Thus,  in  the  case  of  a  one-dimensional  sub-domain  or  line, 


dp       dp 
In  this  case  h  =  hn  and,  therefore, 

dtt=V~hn  dp, 
which  is  ds  itself,  as  it  should  be. 

For  a  two-dimensional  sub-domain  or  surface  we  have 

ds*  =  An  dpi*+2hl2  dpi  dpi+hn  dp?, 

dxt       dxK 
where  hab=giK  -T— -    -r— -  . 

Thus, 

dfi  =  V  A  d/>!  d/>2, 
where 

djct       5jcK  d#t     ^^K  /"  d.rt 

==  f*lK  O  .  «J  .  •  SlK  f\  f\  \          &IK     "     n 

Opi         Opi  Op2     Opz  \  Opl 


COVARIANT  DERIVATIVES  59 

23.  Differentiation  based  on  metrics.  We  have  already  seen 
(p.  49)  that  if  /  be  a  scalar  or  invariant,  df/dxt,  the  gradient 
of/,  is  a  covariant  vector.  This  is  independent  of  the  metrics 
of  the  manifold.  But,  as  was  then  pointed  out,  the  iterated 
application  of  the  operation  d/dxt  would  not  lead  to  tensors; 
nor  would  its  application  to  a  vector  A,  or  another  tensor 
yield  by  itself,  unaided  by  auxiliaries  such  as  glK  ,  a  tensor. 
But  the  introduction  of  the  metrical  tensor  opens  in  this 
respect  new  and  important  possibilities. 

It  was  remarked  by  Christoffel  as  long  ago  as  1869  that 
if  A  t  be  a  covariant  tensor,  so  is 


namely  covariant,  of  rank  two.    Similarly  if  B^  be  a  covariant 
tensor  of  rank  two, 


B..-  j  ^  |  £.. 

is  again  a  covariant  tensor  of  rank  three;  similarly 

*r  (42a) 


is  a  mixed  tensor  of  rank  three,  and  so  on.  But  it  will  be 
enough  to  consider  here  at  some  length  the  first  case  (41) 
only,  especially  as  the  other  cases  can  be  derived  from  it. 
The  operation  indicated  in  (41)  is  called  covariant  differentia- 
tion, and  its  result  AIK  the  covariant  derivative  or  the  expansion 
(Erweiterung)  of  At. 

If  5l  be  a  contra  variant  vector, 


is  a  contravariant  tensor  of  rank  two,  the  contravariant  derivative  of  Z?'. 
But  for  our  purposes  it  will  suffice  to  consider  only  the  covariant 
differentiation. 

That  (41)  represents  a  covariant  tensor  can  be  proved  in 
a  variety  of  ways.    The  most  instructive  of  these  is  perhaps 


—  5 


60  RELATIVITY  AND  GRAVITATION 

that  given  by  Einstein,  since  it  makes  immediate  use  of  the 
equations  of  geodesies,  and  the  r61e  of  the  Christoffel  symbols* 
appearing  in  (41)  is  thus  far  known  to  us  only  in  connection 
with  these  world-lines.  Einstein's  reasoning  is  as  follows: 

Let/  be  a  scalar  or  better  a  scalar  field  (i.e.  an  invariant 
function  of  position  within  the  world).  Differentiate  it  twice 
along  any  world-line.  Then 


d2f  df       d2xt  d2/         dxa 

ds2  dxt      ds2  dxtdxp       ds       ds 

will  again  be  an  invariant.    Let  the  line  be  a  geodesic.   Then 

=    —  \         ( — —  ,  and  the  invariant  will  assume 

ds2  I    <-    )      ds      ds 

the  form 


&L  =  r  a* 

ds*          Ldxa 


dxft  (    «•    )    d#tJ   ds       ds 

Since  the  contravariant  tensor  (of  rank  two)  dxa  dx#  is  arbitrary 
(for  from  a  given  point  a  geodesic  can  be  drawn  in  any  direc- 
tion, i.e.  with  arbitrary  ratios  of  dxit  <fe>  etc.)  and  its  product 
into  the  bracketed  term  is  invariant,  the  latter,  i.e. 

(43) 
dxa 

is  a  covariant  tensor  of  rank  two.  This  proves  the  proposition 
for  the  special  vector  At  —  df/dxt.  To  prove  it  for  any 
covariant  vector,  notice  that  any  such  vector  At  can  be  repre- 
sented by  the  sum  of  four  (generally  n)  terms  of  the  form 
ifrdf/dXt,  where  \f/  and  /  are  scalars.  Thus  it  is  enough  to 
prove  that 


dxK  \       dxt 

is  a  tensor.     But  this  is  equal  to 


*Notice  in  passing  that   <     !  '*        is  not  a  tensor. 


<J    !  '*    i  i 


COVARIANT  DIFFERENTIATION  61 


which,  being  the  sum  of  covariant  tensors  of  rank  two,  is 
itself  a  tensor  of  the  same  kind  and  rank. 

Thus  the  tensor  character  of  the  derivative  (41)  of  any 
vector  A  i  is  proved.  Notice  that  for  constant  glti  (galilean 
world)  the  Christoffel  symbols  vanish  and  this  covariant 
tensor  of  derivatives  reduces  to  an  array  of  ordinary^deriva- 
tives  dAJdxK. 

The  proof  of  the  tensor  character  of  (42),  which  can  be  easily'Meduced 
from  that  of  (41),  may  be  left  to  the  care  of  the  reader.  It  is  interesting  to 
note  that  the  covariant  derivative  of  the  metrical  tensor  g^  itself  vanishes 
dentically.  In  fact,  substituting  in  (42)  glK  for  BIK  we  have 


and  since 


which  will  also  be  useful  in  other  connections,  and  similarly  for  the  last 
term,  we  have 


dglK 


m-m 


But  by  the  definition  (13)  of  the  symbols,  and  since  g^  =g    ,  we  have 

-- 


Thus,  g«x  =  0,  identically. 

Let  AIK  be  as  in  (41),  where  At  stands  for  any  covariant 
vector.  Then,  since  the  second  term  in  (41)  is  symmetrical 
in  i,  K,  dAJdxK  —  dAJdxt  =  Aui  —AKi  ,  being  the  difference  of 
two  tensors,  is  again  a  tensor.  This  tensor  is  called  the  rotation 
of  the  covariant  vector  Alt  and  can  be  written 


J5.  (45) 

dxa 


62  RELATIVITY  AND  GRAVITATION 

This  covariant  tensor  of  rank  two  is  manifestly  antisym- 
metrical,  i.e.,  in  the  case  of  a  four-  manifold,  a  six-vector. 
Notice  that  although  the  proof  of  the  tensor  character  of  the 
rotation  was  based  on  the  metrical  formula  (41),  yet  the 
rotation  itself,  as  denned  by  (45),  is  entirely  independent  of 
the  metrical  properties  impressed  upon  the  manifold.  It 
contains  no  trace  of  the  metrical  tensor  g^  . 

The  same  is  true  of  a  tensor  of  rank  three  which  can  be 
deduced  from  (42).  Let  in  that  formula  BIK  be  an  anti- 
symmetric tensor  or  six-vector.  Then 

R     j_  R    _i_  R          **BIK          dBKX         SB^L  ,     . 

B^+B^+B^^  —  -    +    -    -f  -  .  (46) 

O.TX  oxt  OXK 

Thus  the  right  hand  member  is  again  a  tensor.  This  is  called 
the  antisymmetric  expansion  of  the  six-vector  BiK  .  It  will, 
together  with  the  rotation  (45),  be  of  use  in  connection  with 
electromagnetism. 

Another  tensor  derived  from  a  six-vector  of  equal  import- 
ance in  the  said  connection  is 


(47) 


a  contra  variant  vector,  called  the  divergence  of  the  contravariant 
six-vector  AIK  —  —  AKl  .  The  proof  of  its  tensor  character,  to  be 
based  on  (42),  can  be  omitted  here. 

Finally  let  us  mention,  without  proof,  that 


V^7    "7    (Vg^)  =  div(^«),  (48) 

called  the  divergence  of  the  contravariant  vector  A",  is  a  scalar  or 
invariant. 

24.  The  Riemann-Christoffel  tensor  is  of  such  capital  im- 
portance for  Einstein's  gravitation  theory,  and  for  the 
geometry  of  any  riemannian  manifold,  as  to  deserve  to  be 
treated  at  some  length. 

It  is  a  metrical  tensor  of  rank  four,  built  up  of  g^  and  their 
first  and  second  derivatives,  known  to  the  general  geometers 
since  the  time  of  Riemann. 


THE  CURVATURE  TENSOR  63 

It  expresses  the  so-called  curvature  properties  of  a  mani- 
fold or  w-space  whose  metrical  relations  are  fixed  by  the  tensor 
£„  ,  and  to  Einstein  it  served  as  the  material  for  building  up 
his  gravitational  field-equations. 

In  order  to  arrive  at  this  all-important  tensor  let  us  start 
from  an  arbitrary  co  variant  vector  A,  and  let  us  write  down 
its  second  covariant  derivative,  that  is  to  say  the  covariant 
derivative  of  the  tensor  AtK  which  is  the  covariant  derivative 
of  Alt  i.e.,  by  (42), 


where  AIK    is  as  in  (41).     Similarly,  transposing  K  and  X,  let 
us  write  the  second  covariant  derivative 


dxK 
Either  being  a  third-rank  tensor,  so  will  be  their  difference 

A^-Ai^  8AlK  -    dA* 
This  is,  by  (41), 


In  the  second  term  the  indices  a  and  #  over  which  the  sum  is 
to  be  taken  can  be  interchanged.  Thus  A^  —  A^K  is  the  inner 
product  of  an  arbitrary  covariant  vector  Aa  into  the  sum  of 
the  two  bracketed  expressions.  This  sum,  therefore, 

(49) 


is  a  mixed  tensor  of  rank  four.  This  is  the  Riemann-Christoffel 
tensor  which,  for  reasons  to  appear  presently,  may  as  well  be 
called  the  curvature  tensor. 


64  RELATIVITY  AND  GRAVITATION 

Strictly  speaking,  Riemann's  own  system  of  four-index 
symbols  (iju,  XK),  discovered  in  1861  in  connection  with  a 
problem  in  heat  conduction  (Mathematische  Werke,  2nd  ed., 
p.  391),  is  the  purely  covariant  tensor  associated  with  (49), 
to  wit 

.  (50) 


From  this  we  have  conversely, 

#A  =  g*B(tM,XK).  (50a) 

Also  the  latter  tensor  was  used  in  geometry  for  a  long  time.* 
The  Riemann  symbols  are,  for  an  w-space,  w4  in  number, 
and  for  our  world,  therefore,  as  many  as  256.     But  they  are 
bound  to  one  another  by  the  linear  relations 

(tju,  /cX)  =  —  (/u,   K\),  (tju,  *X)  =  —  (IM>   XK),   (iM>   *cX)  =  (/cX,  i/z), 


so   ttyat   the   number  of  essentially  different,   i.e.    linearly 
independent  symbols  is  reduced  to 


12 

For  a  proof  see,  for  instance,  Killing,  loc.  cit.,  p.  228. 

In  the  case  of  a  one-dimensional  manifold,  a  line,  there 
is  no  such  non  -vanishing  symbol.  In  fact,  although  a  line  may 
be  '  curved  '  from  the  standpoint  of  two-  or  more-dimensional 
beings  in  whose  space  it  is  imbedded,  yet  it  has  no  intrinsic 
properties  of  its  own  to  distinguish  it  from  other  lines,  nor 
one  of  its  parts  from  another.  Take,  for  instance,  a  plane 
curve.  If  Aco  be  the  angle  between  the  tangents  at  two  points 
separated  by  the  arc  As,  the  curvature  of  the  line  is  defined 
as  the  limit  dco/ds.  Now,  this  curvature  is  often  called  an 
intrinsic  property  of  the  line,  because  (unlike  the  sloping 
of  the  line)  it  is  independent  of  a  coordinate  system  laid  in 

*Cf.  for  instance  L.  Bianchi,  1902,  loc.  cit.,  p.  72,  where  it  is  denoted  by 
|  ta,  X/c|.  The  geometrical  applications  of  the  Riemann  symbols  are  fully 
treated  in  vol.  I  of  Bianchi's  work.  See  also  W.  Killing's  Nicht-Euklidische 
Rawnformen,  Leipzig  (Teubner),  1885. 


RIEMANN  SYMBOLS  65 

that  plane,  yet  it  is  entirely  meaningless  if  the  line  is  not 
conceived  as  a  sub-domain  of  the  plane.  For  so  is  the  angle  Aco. 
And  from  the  bidimensional  standpoint  every  curve  is 
developable  upon  every  other. 

In  the  case  of  a  surface,  n  =  2,  there  is,  by  (51),  essentially 
just  one  Riemann  symbol,  namely 

(12,  12), 

(21,  21)  being  equal,  and  (12,  21),  (21,  12)  oppositely  equal 
to  it,  and  all  others  being  zero.  This  unique  symbol  divided 
by  the  discriminant  g  is  a  differential  invariant  of  the  surface 
(or  of  its  metrical  form  ds2>=gndxi2-}-2gttdxidx2-\-g2<idx22).  This 
invariant, 

K  -    (12'  12)     =       (I2'12)     ,  (52) 

g 


is  the  familiar  gaussian  curvature  of  the  surface,  its  reciprocal 
being  the  product  of  the  two  principal  radii  of  curvature. 
This  is  an  intrinsic  metrical  property  of  the  surface,  requiring 
for  its  general  definition  or  its  numerical  evaluation  no  refer- 
ence whatever  to  a  third  dimension.  In  fact,  (52)  contains 
only  the  metrical  tensor  components  glK  and  their  first  and 
second  derivatives  with  respect  to  any  gaussian  coordinate 
system  spread  over  the  surface  itself  as  a  network  of  lines. 
The  curvature  thus  defined,  in  general  variable  from  point  to 
point,  can  be  evaluated  at  any  spot  by  dividing  the  excess 
of  the  angle  sum  (over  a  straight  angle)  of  an  infinitesimal 
triangle  by  the  area  of  the  triangle.  Again,  as  is  well  known, 
the  necessary  and  sufficient  condition  for  a  surface  to  be 
developable  upon  a  euclidean  plane  or  for  its  fundamental 
form  to  be  holonomously  transformable  into 


is  the  vanishing  of  K,  i.e.  of  (12,  12)>,  and  herewith  the  vanish- 
ing of  the  whole  tensor  of  Riemann  symbols. 

For  a  three-space  there  are,  by  (51),  six,  and  for  the  world 
or  space-time  as  many  as  twenty  independent  Riemann 
symbols.  A  five-space  has  fifty  independent  symbols,  and  so 
on.  But,  no  matter  what  the  number  of  dimensions,  the 


66  RELATIVITY  AND  GRAVITATION 

Riemann  symbols  always  represent  the  curvature  relations 
of  the  manifold,  and  their  vanishing  continues  to  form  the  con- 
dition of  an  important  property  of  the  metrical  form  of  the 
manifold. 

To  begin  with  the  latter,  suppose  all  gtK  are  constant  over 
a  domain  of  the  world.  Then  all  (tju,  X/c),  and  therefore  also 
all  the  components  of  the  tensor  B"KX  vanish  throughout  the 
domain.  This  then  is  the  necessary  condition  for  a  domain 
of  the  world  to  be  galilean,  i.e.,  for  the  line-element  to  be 
holonomously  transformable  into  ds2  =  c^dt?  —  dx2  —  dy2  —  dz2. 
It  was  proved  by  Lipschitz  that  this,  i.e. 

£ix  =  0, 

is  also  the  sufficient  condition  for  the  said  reducibility  (to  a 
form  with  constant  coefficients). 

In  the  second  place,  concerning  the  curvature  relations, 
consider  a  surface  or  or  a  two-dimensional  sub-domain  of  the 
world,  or  of  any  metrical  manifold.  More  especially,  let  <r 
be  a  geodesic  surface.  This  can  be  defined  as  follows.  From 
a  point  0(xt)  draw  two  infinitesimal  vectors  d£t,  dr^t  and  con- 
sider the  pencil  of  infinitesimal  vectors 


where  a,  /3  are  free  coefficients.  Draw  from  0  the  geodesic 
lines  with  each  of  these  vectors  as  initial  direction.  The 
surface  thus  constructed  will  be  a  geodesic  surface  through  0. 
Its  normal  v  will  be  defined  by  the  infinitesimal  vector  dyt 
orthogonal  to  d%t  and  drji,  i.e.,  such  that 

giKdyid^K  =  giKdy,drlK  =  0, 

and  therefore  also  glK  dy,dxK  =  Q.  The  geodesic  surface  ff  =  vv 
will  thus  be  completely  determined  by  O,  one  of  its  points, 
and  by  its  orientation,  given  by  the  normal  just  explained. 
The  line-element  of  the  manifold  (world)  will  give  for  the 
line-element  of  this  surface,  at  0,  an  expression  of  the  form 

ds2  =  hn  dpl2+2hl2  dpi  dp2+hn  dp22, 
and  the  gaussian  curvature  of    a,,  will  be,  as  before,  with 


RlEMANNIAN    CURVATURE 


67 


(12,  12), 
h 


(52a) 


This  is,  according  to  Riemann's  definition,  the  curvature  of  the 
manifold,  at  O,  corresponding  to  the  orientation  v  of  the  geodesic 
surface.  The  suffix  h  has  to  remind  us  that  the  Riemann 
symbol  is  to  be  formed  with  hlK  as  the  fundamental  tensor  (of 
the  sub-manifold  <*•„)•  It  remains  to  express  (52o)  in  terms 
of  the  original  tensor  gLK  of  the  manifold  and  the  vectors 
d£t,  drjt  defining  the  orientation  of  the  surface.  This  gives 
ultimately* 


(53) 


where 


-S'(lX,    KfJ.)g 

h 


Sue 


the  dashed  sums  to  be  extended  only  over  such  combinations 
of  the  indices  for  which  i  <X  and  at  the  same  time  K <ju- 

This  will  suffice  to  show  the  role  of  the  four-index  symbols 
in  determining  the  riemannian  curvature  of  a  metrical 
manifold  of  any  number  of  dimensions.  In  general,  the 
curvature  will  not  only  vary  from  point  to  point  but  will 
also  be  different  for  different  surface  orientations  (v).  In 
short,  the  manifold  will  in  general  be  heterogeneous  and 
anisotropic  with  regard  to  its  curvature.  Such,  for  instance, 
will  be  the  world  as  the  seat  of  a  permanent  gravitational  field. 
On  the  other  hand,  a  galilean  domain,  for  which  all  (tX,  KM) 
vanish,  will  have  everywhere  and  for  every  orientation  the 
curvature  zero.  In  other  words,  it  will  be  flat  or  homaloidal. 
The  next  simple  case  is  that  of  a  manifold  of  positive  or 
negative  constant  curvature,  for  which,  that  is,  KV  —  K  is 
constant  and  equal  for  all  directions  of  v.  It  may  be  interesting 
to  note  that  the  necessary  and  sufficient  condition  for  the 
constancy  and  isotropy  of  curvature  is 

*Cf.  Bianchi,  loc.  cit.,  pp.  340-343. 


68  RELATIVITY  AND  GRAVITATION 

(iX,  K/i)=-K(g«  &/»-.&*»&«)»  (54) 

for  all  values  of  the  indices  i,  K,  X,  /*.  These  are  partial 
differential  equations  of  the  second  order  for  the  gtK  with  K 
as  a  constant  coefficient.  They  exhibit  the  flat  manifold, 
.K"  =  0,  as  a  sub-case. 

Other  concepts  connected  with  the  system  of  riemannian 
curvatures  KVJ  such  as  the  mean  curvature,  will  be  given 
later  on  in  connection  with  gravitational  problems.  Here  our 
purpose  was  only  to  show  that  the  curvature  of  the  four- 
world  and,  in  fact,  of  a  metrical  manifold  of  any  number  of 
dimensions,  is  a  concept  as  definite  and  essentially  as  simple 
as  that  of  an  ordinary  surface.  The  only  difference  is  that  of 
a  possible  anisotropy  of  Kv  for  three-  and  more  dimensional 
manifolds,  whereas  there  is,  of  course,  no  such  possibility  in 
the  case  of  a  surface. 

We  are  now  ready  to  explain  the  use  made  by  Einstein  of 
the  Riemann-Christoffel  or  the  curvature-tensor  in  constructing 
his  gravitational  field  equations.  These  will  occupy  our 
attention  in  the  next  chapter. 


CHAPTER  IV. 

The    Gravitational  Field-equations,    and  the   Tensor 

of  Matter. 


25.  As  was  pointed  out  on  several  occasions,  the  funda- 
mental, metrical  tensor  gllf  of  the  world  determines,  through 
the  line-element  ds2  =  glK  dxkdxK,  its  null-lines  and  its  geodesies, 
and  these,  in  virtue  of  the  explained  concrete  representation, 
rule  the  propagation  of  light  and  the  motion  of  a  free  particle, 
respectively.  It  remains  to  build  up  generally  co  variant  laws 
or  equations  which  would  enable  us  to  determine  the  metrical 
tensor  g^  itself.  Needless  to  say  that  in  looking  after  such 
equations  Einstein  had  in  view,  from  the  outset,  the  gravita- 
tional field.  Of  this  he  knew  that  to  a  certain  degree  of 
approximation  it  was  represented  by  the  (non-covariant) 
differential  equation  of  Laplace-Poisson  for  the  ordinary 
potential, 


the  gradient  of  the  potential  ft  giving  the  right  hand  member 
of  Newton's  (approximately  valid)  equations  of  motion.  The 
equation  of  Laplace-Poisson  being  of  the  second  order  it  was 
natural  to  look  for  a  tensor  containing  the  second  deriva- 
tives of  the  metrical  tensor  components  together  with  the 
gM  themselves  and  their  first  derivatives. 

Such  a  tensor  lay  ready  in  the  treasury  of  the  geometry 
of  w-dimensional  spaces  since  the  time  of  Riemann,  and  it 
represented,  moreover,  certain  intrinsic  properties  of  any 
such  metrical  manifold,  its  curvature  properties.  This  was 
the  covariant  tensor  of  the  four-index  symbols  (t/i,  XK)  or  the 
associated  mixed  Riemann-Christoffel  Censor  .B*x.  It  was 
natural,  therefore,  and  Einstein  himself  relates  to  us  that 
such  was  his  first  thought,  to  utilize  for  the  purpose  in  hand 
this  very  tensor. 

69 


70  RELATIVITY  AND  GRAVITATION 

As  we  saw  in  the  last  chapter,  the  vanishing  of  this  tensor 
expressed  a  simple  and  at  the  same  time  a  profound  feature 
of  a  metrical  manifold,  to  wit,  the  ~  n*(n?  —  1)  independent 
equations 

A"x  =  0 

formed  the  sufficient  and  necessary  condition  for  the  flatness 
of  the  manifold.  In  our  case  twenty  such  equations  form 
the  necessary  and  sufficient  condition  for  a  world-domain  to 
be  essentially  galilean,  i.e.,  for  its  line-element  to  be  holo- 
nomously  transformable  into  c^fi  —  dx^—dy^  —  dz2. 

A  domain,  therefore,  in  which  there  is  no  gravitational 
field,  i.e.,  no  premanent  field  of  acceleration,  will  certainly 
be  characterized  by  the  generally  covariant  equations  .S*x  =  0. 
The  same  equations  might  at  first  suggest  themselves  for  the 
description  of  a  gravitational  field  outside  of  matter.  It  will 
be  seen,  however,  after  a  moment's  reflection  that  they  would 
be  too  stringent  for  such  purposes.  In  fact,  the  field  of 
acceleration  surrounding  the  sun,  say,  can  certainly  not  be 
transformed  away  holonomously.  The  said  equations  would 
thus  be  too  stringent  for  such  fields  and,  in  fact,  for  any 
acceleration  field  which,  in  our  nomenclature,  is  a  permanent 
field,  i.e.,  not  to  be  got  rid  of  by  any  holonomous  transfor- 
mations of  the  coordinates.  At  the  same  time,  the  g^  to  be 
determined  are  only  ten  in  number,  forming  a  symmetrical 
tensor  of  rank  two,  while  the  Riemann-ChristorTel  tensor  is  of 
rank  four,  and  consists  of  twenty  independent  components. 
Such  considerations  led  Einstein  to  require  for  the  gravita- 
tional field  outside  of  matter  a  set  of  broader  equations, 
yet  of  the  second  order,  and  ten  in  number.  For  this  purpose 
the  symmetrical  tensor  derived  from  J34aKX  by  contraction  with 
respect  to  a,  X  naturally  suggested  itself. 

In  fine,  writing  B"Ka  =  GIK  ,  Einstein's  field  equations  outside 
of  matter  are 

G,  =  0.  (Ill0) 

That  G^  ,  obtained  by  contraction  from  the  mixed  tensor 
of  rank  four,  is  itself  a  covariant  tensor  of  rank  two,  we  know 


FIELD  EQUATIONS  71 

from  the  preceding  chapter.     Moreover,  by  (49),  to  be  con- 
tracted with  respect  to  X,  a,  we  have 

(55) 


or,  after   some   simple   transformations   which    can    here   be 
omitted, 


5    _ 


_  d2logV-g 


J  IK 
I  a 


dx 


Now,    <l  >,  so  that  SIK  (which  is  not  a  tensor) 

is  symmetrical,  and  such  being  also  the  first  two  terms  in 
(55o),  GM  is  seen  to  be  a  symmetrical  covariant  tensor,  of 
rank  'two ;  GIK  =  GKl  . 

Thus  Einstein's  field  equations  (III0),  valid  outside  of 
matter,  are  ten  in  number,  and  such  is  exactly  the  number  of 
the  metrical  tensor  components  gllc  .  The  field  equations 
would  then  give  us  a  system  of  ten  differential  equations  of 
the  second  order  for  ten  unknown  functions  gllc  of  the  co- 
ordinates. As  a  matter  of  fact,  however,  there  exist  between 
the  covariant  derivatives  Gt/cX  of  the  GIK  and  the  derivatives 
3G/dxL  of  the  invariant  G  =  glK  GIK  four  identical  relations 
(based  upon  certain  identical  differential  relations  discovered 
by  Bianchi),  to  wit 

G,  =  g"x  G^  =  \  -  -  ,  t  =  1,  2,  3,  4.  (56) 

dxt 

Owing  to  these  four  identities,  to  which  we  shall  have  to 
return  later  on,  only  six  of  the  field  equations  are  mutually 
independent,  leaving  therefore  four  of  the  g^  or  any  four 
functions  of  the  glK  free  or  undetermined.  Such,  however, 
should  from  the  general  relativistic  standpoint  be  the  case. 


72  RELATIVITY  AND  GRAVITATION 

In  fact,  from  this  point  of  view  one  would  expect  beforehand 
the  field  equations  or  any  differential  laws  to  be  such  as  to 
leave  us  a  perfectly  free  choice  of  the  system  of  coordiantes. 
Einstein  himself,  for  instance,  makes  use  of  this  freedom  by 
putting  in  most  of  his  formulae  V—  g  =  l,  which,  by  (55a), 
reduces  his  field  equations  to 


and  leaves  him  still  a  threefold  freedom  of  choice.  The  latter 
can  often  be  used  with  advantage  by  making  g14  =  g24  =  g34  =  0. 
It  will  be  kept  in  mind,  however,  that  the  equations,  such  as 
(57),  thus  simplified  do  not  retain  their  form  under  general 
transformations.  They  are  only  useful  as  technical  devices 
offering  some  advanatges  in  the  treatment  of  special  problems. 
The  generally  covariant  form  of  the  field  equations  is  only  that 
obtained  by  equating  to  zero  the  complete  or  general  value  of 
G^  ,  such  as  (55)  or  (55a). 

26.  In  order  to  see  the  relation  of  Einstein's  field  equa- 
tions to  the  more  familiar  Laplace  equation,  let  us  evaluate 
the  curvature  tensor  GIK  for  the  case  of  a  'weak'  field,  i.e. 
differing  but  litle  from  a  galilean  domain.* 

Thus,  using  a  quasi-cartesian  system  of  coordinates,  let 
the  fundamental  tensor  differ  but  little  from  the  galilean 
tensor  glK  ,  i.e.,  as  in  (21),  let 

g«  =  g«  +  7«K  , 

where  all  the  ytlc  are  small  fractions.  Then  the  products  of  the 
Christoffel  symbols  in  (55)  will  be  small  of  the  second  order, 
and  the  tensor  in  question  will  be  reduced  to 


dxK      <*  dxa 

Here,  up  to  second  order  terms, 


*Notice  in  passing  that  all  gravitational  fields  known  from  experience 
are  'weak'  in  this  sense  of  the  word. 


FIELD  EQUATIONS  73 

and  since  ^n  =  g22  =  !:33=  —  1,  1*44=  1,  while  all  other  "g"  vanish, 
(LK\  rtK~l 

<      .     >    =  -  .         .  *  =  1,  2,  3; 

\*  }        L  i  J 


Thus,  using  the  index  i  for  1,  2,  3  and  summing  every 
term  in  which  i  occurs  twice  over  1,  2,  3,  we  have  the  approxi- 
mate curvature  tensor 


In  the  present  connection  the  only  interesting  component  is 
that  corresponding  to  i  =  K  =  4.  This  is,  by  (58),  and  on  sub- 
stituting the  values  (13)  for  the  Christoffel  symbols, 

(680) 


oXi  0X4          oxf 

a2  .  a2       a2       a2 

where  V2  =  -  is  the  well-known  Laplacian  --  +  --  h  —  • 
dxf  dxi2      a*22      d*s2 

If  the  field  is  stationary,  the  second  and  the  third  terms 
vanish  and  Einstein's  last  field  equation,  6^4  =  0,  reduces  to 
the  familiar  equation  of  Laplace 

V?£44  =  0.  (59) 

At  the  same  time,  as  we  saw  before  (p.  36),  the  equations 
of  motion  assume,  in  absence  of  #4,-,  the  form  of  Newton's 
equations 


j«     =   -IT'  (236) 

dt2  dXi 

c2 
where  the  potential  fi  =  -  —  £44,  differing  only  by  a  constant 

factor  from  £44,  again  satisfies  Laplace's  equation.  The 
complete  contents  of  Newton's  law  of  gravitation,  thus  far 
outside  of  matter,  appear  as  a  first  approximation  to  Einstein's 
field  equations  and  his  equations  of  motion  of  a  free  particle. 


74  RELATIVITY  AND  GRAVITATION 

27.  The  ten  field  equations  G^  =  0  are  valid  outside  of 
'matter',  i.e.,  as  is  expressly  stated  by  Einstein,  in  such 
domains  of  space-time  in  which  there  is  not  only  no  matter 
in  the  ordinary  sense  of  the  word  but  also  no  electromagnetic 
field,  or,  in  fact,  no  distribution  of  energy  of  any  origin  other 
than  gravitational.  Following  Einstein's  example  the  word 
'matter'  will  be  used  to  cover  all  such  cases.  This  will  har- 
monise with  the  property  of  energy  already  familiar  to  us 
from  special  relativity,*  namely  of  possessing  inertia,  an 
amount  of  energy  U  being  equivalent  to  an  inert  mass  U/c~t 
which,  by  the  law  of  proportionality,  is  also  its  heavy  or 
gravitational  mass. 

As  we  saw  before,  the  role  of  the  newtonian  gravitation 
potential  12  is,  in  a  first  approximation,  taken  over  by  the 
tensor  component  g44  multiplied  by  —  c2/2.  The  vanishing  of 
G44  was  approximately  equivalent  to  Laplace's  equation 
V212  =  0  which  holds  outside  of  matter.  Within  matter 
Laplace's  equation  is  replaced  in  the  classical  theory  of 
gravitation  by  the  more  general  equation  of  Laplace-Poisson, 


where  p  is  the  density  of  mass  in  astronomical  units.  f    Now, 
since   G44  reduces   approximately,   in   a   stationary   field,    to 

—  iV2g44==   —  V2fl,  the  idea  easily  suggests  itself  to  make 


,  (60) 

0 

and  to  consider  this  as  the  equation  or  at  least  as  one  of  the 
field  equations  within  matter.  But,  needless  to  say,  such  a 
single  equation  would  not  by  itself  serve  any  relativistic 
purpose.  What  is  required  is  a  system  of  ten  equations,  of 


*And  partly  even  from  pre-relativistic  considerations,  such  as  in 
Mosengeil's  investigations  on  an  enclosure  filled  with  radiation  or  those 
made  in  connection  with  Poynting's  light-pressure  experiments. 

fit  will  be  kept  in  mind  that  a  mass  m  in  astronomical  units  is  defined 

by  =  force,  so  that  its  dimensions  are 

\m]  =  [length  X  (velocity)2]. 


TENSOR  OF  MATTER 


75 


which  this  should  be  one.  In  other  words,  the  tensor  GIK  has 
to  be  made  equal  or  proportional  to  a  symmetrical  co- 
variant  tensor  of  rank  two  somehow  associated  with  'matter' 
and  having  for  its  44-component  the  density  p  or  what  ap- 
proximately reduces  to  the  usual  mass  density  and  therefore, 
apart  from  a  constant  factor,  to  energy  density.  Now,  such 
a  tensor  was  familiar  from  the  special  relativity  theory  under 
the  name  of  stress-energy  tensor  often  abbreviated  to  energy 
tensor.  The  merit  of  having  introduced  this  concept  into 
modern  physics  is  chiefly  due  to  Minkowski  and  Laue,  pre- 
ceded in  non-ielativistic  physics  by  Max  Abraham.  The 
energy  tensor  made  its  first  appearance  in  electromagnetism, 
in  connection  with  the  ponderomotive  properties  of  an  electro- 
magnetic field,*  as  the  symmetrical  array  or  matrix 


/ll  /12  /13  Pi 
/21  /22  /23  p2 
/31  /32  /33  p3 
Pi  Pi  pz  —  U 


f         P 

p  -u 


consisting  of  the  six  components  fik  =/£,•  of  the  maxwellian 
electromagnetic  stress,  of  twice  the  three  components  pi  of 
electromagnetic  momentum  (or  Poynting's  energy  flux)  and 
of  the  density  u  of  electromagnetic  energy.  The  physical 
significance  of  this  tensor  or  matrix  was  that  its  product  into 
the  operational  matrix 


lor 


dx 


dy         dz 


dt 


gave  the  ponderomotive  force  P  per  unit  volume  and  its 
activity  Pv, 

-lorS  =  |PltP2,P8,Pv|. 

Later  on  its  r61e  was  generalized  for  a  stress,  momentum  and 
energy  density  of  any  origin,  not  necessarily  electromagnetic, 

*Cf.  Theory  of  Relativity,  Macmillan,  1914,  Chap.  IX,  especially  p.  238. 
In  reproducing  it  here,  with  pi  written  for  gi  f  I  drop  the  imaginary  unit 
and  put  c  =  l. 


76  RELATIVITY  AND  GRAVITATION 

provided  only  that  the  force  and  its  activity  could  be  repre- 
sented in  the  form 


dt  dt 

From  the  special  relativistic  standpoint  this  array  of  ap- 
parently heterogeneous  physical  magnitudes  was  important 
as  it  transformed  from  one  inertial  system  S  to  another  Sr  as 
a  whole,  to  wit  by  the  operator  A(  )A,  where  A  is  the  funda- 
mental Lorentz  transformation  matrix  of  4X4  elements  and 
A  the  transposed  of  A.  The  developed  form  of  the  trans- 
formation equations  of  stress-momentum-energy  need  not 
detain  us  here.* 

The  important  thing  in  our  present  connection  is  that  the 
said  stress-momentum-energy  array  is  a  symmetrical  tensor 
of  rank  two.  And  since  such  also  is  the  contracted  curvature 
tensor  GtK  ,  the  idea  naturally  suggests  itself  to  make  GIK  pro- 
portional to  a  symmetrical  covariant  tensor  TtK  ,  of  which  the 
first  nine  components  7\i,  T\z,  .  .  .  T^  are  of  the  nature  of 
stress  or  equivalent  to  it,  the  components  2V(*=i,  2,  3)  replace 
the  momentum,  and  the  last  component  T^  is,  or  approxi- 
mately reduces  to,  an  energy-  or  mass-density.  But  then  it 
is  by  no  means  necessary  (nor  is  it  possible)  to  fix  beforehand 
the  exact  physical  meaning  of  the  several  components  of  such 
an  energy  tensor  or  tensor  of  matter  (as  it  is  often  called  by 
Einstein).  Their  significance  has  to  be  fixed  a  posteriori, 
through  physical  applications  of  the  field  equations  aimed  at. 

If  TIK  is  a  covariant  tensor  of  rank  two,  then,  as  we  already 
know, 

T  =  f  Tx  (01) 

is  a  scalar,  the  invariant  of  7\K  .f  Such  being  the  case,  glK  T 
is  again  a  symmetrical  covariant  tensor.  Now,  guided  partly 
by  guesses  (originally  at  least)  and  partly  by  considerations 

*It  will  be  found  on  p.  236  of  my  book  quoted  above. 
tSuch  also  was  Laue's  'scalar'  in  relation  to  his   Welttensor',  i.e.  the 
matrix  S. 


FIELD  EQUATIONS  77 

of  conservation  of  energy  and  of  momentum,*  Einstein  wrote 
down  as  his  general  field-equations 

GIK  =  -     ^    (T«  -i&.rj,  (III) 

G 

the  factor  —  87T/C2  being  so  chosen  as  to  give,  in  a  first  approxi- 
mation, the  equation  of  Laplace-Poisson. 

In  fact,  as  we  shall  see  from  the  more  definite  form  to  be  given  pre- 
sently, in  a  first  approximation, 

r44=:r=p, 

4?r 
so  that  the  last  of  (III)  gives  GU= —  p,  as  in  (60).      Einstein's  own 

coefficient  differs  from  ours  by  the  gravitation  constant  which  is  here  in- 
corporated into  p,  the  density  in  astronomical  units. 

The  previous  equations  (III0),  holding  outside  of  matter, 
are  a  special  case  of  these  general  equations,  for  TtK  =  0,  when 
also  r  =  0. 

To  be  exact,  Einstein  speaks  first  of '  matter '  as  'everything 
except  the  gravitation  field'  (loc.  cit.,  p.  802)  and  writes 
GIK  =0  outside  of  matter  in  this  sense  of  the  word.  But  later 
on  (p.  808),  trying  to  justify  the  exact  form  (III)  of  his  general 
equations,  he  states  expressly  that  '  the  energy  of  the  gravita- 
tion field'  (if  there  is  such  a  thing)  has  also  to  'act  gravita- 
tionally  as  every  energy  of  any  other  kind',  in  short  that 
gravitation  energy  too  has  mass  and  weight.  Thus,  rigorously 
speaking,  there  is  'matter'  everywhere,  and  the  equations 
(III0)  are  valid  nowhere,  unless  there  is  no  gravitation  field, 
when  they  are  superfluous.  In  other  words,  gravitation  itself 
contributes  also  to  the  tensor  TIK  .  Its  contribution,  however, 
is  practically  evanescent,  and  this  circumstance  makes  the 
equations  (III0)  physically  applicable. 

But  even  the  contribution  to  Tik  a,k=i,  2, 3)  of  stresses 
within  matter  in  the  ordinary  sense  of  the  word  (tensions  or 
pressures)  is  practically  negligible,  and  so  is  the  contribution 
to  7*44  of  the  energy  proper  outside  of  molecules,  atoms  or 
electrons,  and  we  may  as  well  omit  it  in  7"44,  and  take,  for  a 
first  approximation  at  least,  T44  =  p,  where  p  is  the  density 

*Principles  to  which  we  may  return  later  on. 


78  RELATIVITY  AND  GRAVITATION 

of  ordinary  matter  or  approximately  so,  always  in  astro- 
nomical units.  Thus  the  idea  easily  suggests  itself  to  build 
up  the  tensor  TtK  for  a  theoretically  continuous  body  (a  fluid, 
liquid  or  solid)  out  of  its  local  density  and  the  velocity  com- 
ponents of  its  motion.  For  although  the  gravitational  effects 
of  the  motion  of  matter  are  exceedingly  small,  yet  the  mere 
desire  of  writing  generally  covariant  equations,  say,  of  hydro- 
dynamics,* prevents  us  from  discarding  velocities  in  this 
connection.  Thus,  neglecting  stresses,  etc.,  let  us  introduce, 
after  Einstein,  the  scalar  or  invariant  p  as  'the  density'  of 

matter,  and  the  four-vector  of  velocity    — ' .    Then  p  — "  — * 

ds  ds    ds 

will  be  a  tensor  of  rank  two,  a  contra  variant  one,  however. 
Construct  therefore,  by  the  principles  explained  in  Chapter  III, 
the  associated  tensor 

dxa     dxp 

TiK  =  P  gta  gKp  —     —f-  ,  (62) 

ds      ds 

which  will  be  covariant  and,  manifestly,  a  symmetrical  tensor. 
This  is  Einstein's  energy-tensor  or  tensor  of  matter  to  be 
used  in  (III)  whenever  tensions,  pressures,  etc.,  are  negligible. 
As  a  matter  of  fact,  in  view  of  the  limitations  of  even  the 
most  accurate  methods  of  observations  now  available,  this 
particular  tensor  will  cover,  presumably  for  many  years  to 
come,  all  needs  of  the  physicist  and  the  astronomer  with 
regard  to  gravitation. f 

The  value  of  the  invariant  T  or  the  scalar  belonging  to  this 
tensor,  which  is  defined  by  (61),  follows  at  once.  Since 
f  £.£*  =£&*  =£&*  =g*  ,  and  gapdxa  dxft=dst,  we  have,  by 
(62), 

r=p,  (62a) 

that  is  to  say,  the  scalar  of  the  tensor  in  question  is  the  density 
of  matter. 


*And  this  was  undoubtedly  one  of  the  reasons  by  which  Einstein  was 
influenced. 

•jThe  case  of  hydrodynamics  will  be  covered  by  subtracting  from  (62) 
the  tensor  pglK  ,  where  p  is  the  (invariant)  hydrostatic  pressure. 


FIELD  EQUATIONS  79 

On  the  other  hand,  in  a  local  rest-system,  in  which  only 
i/ds)2  survives  and  is  equal  to  1/gu,  we  have 


£44 

£.e.,  for  instance,  r44  =  pg44,  and  therefore,  by  (III),  rigorously, 

47T 

Cz44=  --  —  g44  P- 
0 

In  a  first  approximation  (g44=rl)  this  gives  (60)  or  the  Laplace- 
Poisson  equation,  as  announced  before. 

The  general  field  equations  (III)  may,  independently  of 
(62),  be  given  a  slightly  different  form.  Multiply  both  sides, 
innerly,  by  gllc  ,  and  write 

G  =  glK  GIK  .  (63) 

Then,  since  g™  gtK  =  4, 

G=  ^L.T.  (64) 

0 

Substitute  this  value  of  T  into  (III).    Then 

C-  -J&.G--  ^frK,  (Ilia) 

C2 

the  required  form  of  the  field  equations. 

Notice  that  G,  as  defined  by  (63),  is  the  invariant  of  the 
curvature  tensor  G^  .  This  invariant  or  rather  one-sixth  of  it 
is  called  the  mean  curvature  of  the  world,  at  the  world-point  in 
question.  In  fact,  in  the  case  of  a  three-space  G  would,  apart 
from  a  mere  numerical  factor,  be  the  arithmetical  mean  of 
the  three  principal  riemannian  curvatures,  and  this  would 
still  be  the  case  for  a  manifold  of  any  number  of  dimensions, 
at  least  if  ds2  be  a  definite,  positive  quadratic  form.  This 
justifies  the  name  given  to  G  above,*  and  equation  (64), 
independent  of  the  particular  form  (62)  of  TIK  ,  teaches  us  that 

*For  a  certain  special  world  to  be  treated  later  on  G  will  be  proved 
explicitly  to  be  six  times  the  smallest  value  of  the  (constant  and  isotropic) 
curvature  of  three-space  which  it  is  possible  to  choose  as  a  section  of  that 
four-world.  (Cf.  Appendix,  A.) 


80  RELATIVITY  AND  GRAVITATION 

this  mean  curvature  is  proportional  to  the  scalar  of  the  tensor 
of  matter  and  vanishes,  therefore,  outside  of  matter.  More 
especially,  if  stresses,  etc.,  be  negligible,  the  tensor  (62) 
conies  to  its  right  and  we  have,  by  (62a)  and  (64), 

G  =  ~  P.  (626) 

The  mean  curvature  of  the  world  is  thus  proportional  to  the 
density  of  matter. 

Notice  that  p/c2  has  the  dimensions  of  a  reciprocal  area, 
and  such  also  are  the  dimensions  of  G,  and  of  all  GIK  ,  since 
the  gtK  are  dimensionless,  and  the  Gllc  are  linear  in  the  second 
derivatives  of  the  glK  with  respect  to  the  coordinates,  each  of 
which  is  a  length.  The  same  remarks  hold  good  with  respect 
to  the  field  equations  (III). 

It  may  be  interesting  to  notice,  even  at  this  stage,  that 
the  mean  curvature  in  familiar  matter,  say,  in  water  under 
normal  conditions,  is,  comparatively  speaking,  not  insignifi- 
cant. In  fact,  remembering  that  the  gravitation  constant 
is  6'658.10~8,  in  c.g.s.  units,  we  have  for  water  at  normal 
density 

G  =     S"    6-658.10-8=r86.1(T27ciri.-2 
9.1020 

What  is  technically  called  the  world-curvature  is  one-sixth 
of  this.  Thus  the  radius  of  mean  curvature  defined  by 
R=  V6/G  will  be,  for  water, 

^  =  5'688.1013cm., 

i.e.,  about  570  million  kilometers  or  only  3'8  astronomical 
length  units. 

But  it  would  be  rash  to  conclude  with  Eddington*  that  a  globe  of  water 
of  about  this  radius  'and  no  larger,  could  exist'.  In  fact,  what  is  known 
from  geometry  is  only  that  the  total  length  of  every  straight  (closed)  tine 
in  a  three-space  of  constant  and  isotropic  curvature  1/-R2,  of  the  properly 
elliptic  or  polar  kind,  is  irR,  so  that  the  greatest  distance  possible  in  such 
a  space  is  fyrR  and  the  total  volume  of  the  space  is  7T2jR3.  But,  unlike  such 

*A  S.  Eddington,  Space  Time  and  Gravitation,  Cambridge,  1920,  p.  148. 


CURVATURE  INVARIANT  81 

a  space,  the  world  has  a  non-definite  fundamental  differential  form,  and 
its  riemannian  curvature  depends  upon  the  orientation  of  the  geodesic 
surface  element.  Thus  a  direct  transfer  of  the  properties  of  an  elliptic 
space  upon  the  (watery)  world  is  certainly  illegitimate.  Notice,  moreover, 
that  G,  and  therefore  R,  is  remarkable  as  a  genuine  invariant  of  the  four- 
world  and  not  of  a  three-space  laid  across  it  as  one  of  an  infinity  of  possible 
sections.  The  best  plan  for  the  present  is,  therefore,  to  see  in  it  only  such 
an  invariant  of  space-time,  within  the  world-tube  of  a  mass  of  water.  The 
few  numbers  were  here  presented  only  to  give  an  idea  of  the  order  of 
magnitude  attainable  by  the  curvature  invariant  in  ordinary  matter,  con- 
sidered as  continuous.  To  those  who  like  to  contemplate  sensational  results 
the  best  opportunity  is  perhaps  afforded  by  the  atomic  nuclei.  According 
to  Rutherford  the  radius  of  the  nucleus  of  the  hydrogen  atom  is  about  one- 
two  thousandth  of  that  of  the  electron,  i.e,  |10-16  cm.,  and  its  mass  practi- 
cally equal  to  that  of  the  whole  atom.  This  would  give  for  the  density  of 
mass  a  value  3.1024  times  the  normal  water  density,  and  therefore  a  curva- 
ture radius  within  the  nucleus  V3  .  1012  smaller,  i.e.,  R  =  32  cm.  only! 
The  moral  would  then  be  that  nuclei  of  about  this  radius  and  no  larger 
could  exist,  with  the  same  density.  Fortunately  they  are  believed  to  be 
much  smaller. 

But  it  is  time  to  return  to  Einstein's  equations  of  the 
gravitational  field  in  order  to  see  some  of  their  further  pro- 
perties. 

28.  Multiply  the  field  equations  (Ilia),  identical  with 
(III),  by  g"a  .  Then,  denoting  by  Gta  and  T?  the  mixed  tensors 
associated  with  GIK  and  TIK  ,  i.e.,  writing 


and   remembering  that  g"a  giK  =  g?  =  d"  ,  we  shall  have,  with 


If  G,ax  be  the  covariant  derivative  of  Gf,  and  similarly  for  the 
energy  tensor,  we  have,  contracting  with  respect  to  X  =  a,  and 
remembering  the  meaning  of  5", 


t 

dxa  dxt 

But,  by  (56),  the  right  hand  member  vanishes  identically. 


82  RELATIVITY  AND  GRAVITATION 

Thus,  as  a  consequence  of  the  field  equations  we  have  the 
four  equations 

^=0  (,  =  1,2,3,4),  (65) 

concerning  the  energy  tensor  or  the  tensor  of  matter.  Thus 
also,  out  of  the  ten  field  equations  only  six  are  left  for  the 
determination  of  the  potentials  gllc  ,  as  announced  before. 

The  matter-equations,  so  to  call  (65),  as  a  consequence 
of  the  field  equations  constitute  a  most  remarkable  result.* 
Notice  that  they  are  entirely  independent  of  any  special  form 
of  the  energy  tensor,  such  as  (62).  They  are  manifestly 
general,  i.e.,  valid  for  any  covariant  tensor  TIK  ,  merely  in 
virtue  of  putting  it  equal,  or  proportional,  to  the  curvature 
tensor 

GIK  -ig«G, 

the  left  hand  member  of  Einstein's  equations. 

In  order  to  see  the  significance  of  the  equations  (65) 
remember  that  T?a  is  the  contracted  covariant  derivative  of 
the  tensor 

Ta  =  <JKO-  T 

-1  t         £       -1  IK  • 

Thus,  if  pressures,  etc.,  are  neglected,  when  TIK  has  the  value 
(62),  we  have,  remembering  that  g*a  &a  g^  =&  gKp  =glf}  , 

dxa     dxa 
7^   —  ^. 
as      as 

More  generally,  if  we  add  to  (62)  a  tensor  pllc  due  to  stresses 
and  any  agents  other  than  molar  motion  of  matter,  we  shall 
have 

(66) 


as       ds 

where  p?=gKa  plK  is  the  associated  supplementary  tensor  of 
matter,  and 


*Of  course,  the  left  hand  members  of  the  field  equations  were  chosen 
by  Einstein  so  as  to  yield  these  four  equations  of  matter,  as  can  be  seen 
by  comparing  his  earlier  paper,  Berlin  Academy,  1915,  pp.  778,  799,  with 
a  later,  improved  paper,  ibidem,  p.  844. 


MATTER-  EQUATIONS  83 

the  co  variant  conjugate  of  the  contravariant  vector  dxa. 
Of  the  mixed  tensor  (66)  we  have  to  take  the  covariant 
derivative,  as  denned  in  (42a),  and  to  contract  it  with  respect 
to  a  and  the  new  index.  Thus  the  equations  (65)  will  be 


.  _    <  „ 

'   dX,          \0)f       \«r 
But,  as  can  easily  be  proved, 

*^L,       ,        '  (67) 


where  g  is  the  determinant  of  the  metrical  tensor. 

Thus,  the  four  equations  (65)  become,  in  any  system  of 
coordinates, 

4=  —  (V^T?)  -(l~\Ti-  0-     <•=>•  *•  »•  *>         (66a) 
V  ff     dxa  \  P  ) 

o 

where  the  mixed  tensor  of  matter  T*  is  as  in  (66).  The  left 
hand  member  of  (65a)  is  a  four-  vector 

T  =  Ta 

•*•  i        •*•  ta 

which  is  called  the  divergence  of  the  mixed  tensor  T*.  The 
equations  (65a)  can  thus  be  read  technically: 

The  divergence  of  the  mixed  tensor  of  matter  vanishes. 

This,  of  course,  does  not  enlighten  us  as  to  their  signifi  - 
cance.  To  see  their  physical  meaning  take  any  coordinate 
system  for  which  g=  —  1,  so  that 


TS  ,  (656) 


and  consider  the  case  of  a  weak  gravitational  field,  for  which 
the  glK  differ  but  little  from  the  galilean  values,  i.e.,  in  quasi- 
cartesian  coordinates,  gn=—  I+TH,  etc.,  as  in  (21).  In  the 
expressions  (66)  for  the  energy  tensor  itself  the  ylK  can  be 
neglected  altogether,  so  that 


84  RELATIVITY  AND  GRAVITATION 


and 


ra_,a,         ^4      dxa 

-I  4     -/>4    +P     —  —  -     —  ~  • 

•  as       as 

In  the  right  hand  member  of  (656)  the  ylK  cannot  be  disre- 
garded without  anihilating  that  member  altogether.  For  the 
Christoffel  symbols  vanish  for  constant  gllc  .  But  since  their 
values  are  taken  to  be  small  of  the  first  order,  it  is  enough  to 
retain  in  the  right  hand  member  only 


and  since    {  l|  )  ==  [4|  ]  =  |  Jfii  ,  the  four  equations  become, 

dxt 

if  pf  be  negligible  in  presence  of  p, 


dxa  dxt 

For  small  velocities,  ds  —  dx4i  =  cdt,  and  if  »,-  be  the  cartesian 
velocity  components  dxi/dt, 


f  etc.  ;  TS  =  ^  -     -   etc.  , 

C 


and 


c  c 

Thus,  neglecting  cpf  and  cp±  in  presence  of  the  momentum 
(per  unit  volume)  pV  of  molar  motion  of  matter,  the  first  three 
equations  will  be 

~  (p^-^H  —  (puM-tpf)  +...  +  -  (pvi)  =P  -,  etc., 
dxi  dx2  dt  dX 

where  12=  —  ic2g44  plays  again   the  part  of  the   newtonian 


MATTER-  EQUATIONS  85 

potential,  and  the  fourth  equation 


d/  dt 

or,  in  obvious  three-dimensional  vector  notation, 


a*  c2  dt 

and,  with  fi  written  for  the  three-vector  C2(pil,  pi2,  pi*), 

•  —  (pvi)  4-  —  (pur)  +         (pvi  z>2)  +  -  -  •  (pvi  v3)  =  div  fi+p  —  ,  etc. 
dt  dxi  8x2  dxs  dxi 

The  left  hand  member  of  the  last  written  equation  is  equal  to 

n  *\  *\ 

—  (pv\)  +^i  --  (pvi)  +  .  .  .  +vs  — 
dt  dXi  dx 

(pvi)+pvi  .  div  V, 
dt 

or,  if  dr  be  the  volume  and  dm  =  pdr  the  mass  of  an  individual 

element  of  matter,  equal  to    —  -  -  .     Similarly  we  have 

dt.  dr 

dp    .    ,.    ,     ,        dp  ,.  d(dm) 

-  +d^v  (Pv)  =  —  -  +pdw  V=  f  —  -  . 
dt  dt  dt  .  6r 

Ultimately,  therefore,  the  approximate  equations  of  matter 
are,  with^i,  j,  k  as  unit  vectors  along  the  coordinate  axes, 


and 


=5m  .  V12  (A) 

dt 


d    /t>    N  dfl    dm 

—  (8m)  =  —  —   —  .  (B) 

dt  dt     c2 


The  first  three  equations  embodied  in  the  vector  formula  (A) 
are  the  equations  of  motion  of  a  continuous  medium*  under 
internal  stresses  (tensions)  fik  =0^,  and  under  the  action  of 

*A  deformable  solid,  liquid  or  fluid. 


86  RELATIVITY  AND  GRAVITATION 

the  gravitational  field  of  which  the  newtonian  potential  is 
again,  as  in  the  case  of  the  approximate  equations  of  motion 
of  a  particle,  12  =  —\&g&.  The  fourth  equation,  (B),  is,  apart 
from  the  new  term  on  the  right  hand,  the  familiar  equation  of 
continuity. 

In  other  words,  the  first  three  equations  express  the 
principle  of  momentum, — the  amount  of  momentum  acquired 
by  matter  from  the  field  per  unit  time  being  given  by 
8m  .  grad  12  which  is  the  newtonian  force  on  the  mass  element 
dm.  And  the  fourth  equation  expresses  the  principle  of  energy 
or,  equivalently,  of  matter, — the  amount  of  energy  (c25m) 
acquired  by  a  material  element  per  unit  time  being  equal 
to  the  mass  8m  of  that  element  multiplied  by  the  local  time- 
variation  of  the  potential,  i.e.,  approximately  equal  to  the 
decrease  of  the  potential  energy  of  that  element. 

It  is  scarcely  necessary  to  say  that  this  gain  or  loss  in  energy  or  in  mass 
of  'matter'  placed  in  a  gravitational  field,  according  to  the  sign  of  d!2/d/, 
is  immeasurably  small.  Its  discussion  in  this  place  may  have  only  a  mere 
academic  interest.  If  it  be  neglected,  (B)  gives  at  once  the  usual  equation 
of  continuity,  and  (A)  assumes  the  perfectly  familiar  form 

pV— i  divfi—  .  .  .  =pV  12  =  gravitational  force  per  unit  volume. 
On  the  other  hand  it  is  interesting  to  notice  that  the  equation  (B)  becomes, 
for  v  =  0,  at  once  integrable  and  gives 

8m  =  8m0  e~^/c   » 

where  8m0  is  8m  for  12  =  0.  A  similar  result  followed  from  a  gravitation 
theory  proposed  some  time  ago  by  Nordstrom  (Phys.  Zeits.,  1912,  p.  11261. 
Its  interpretation  may  be  left  to  the  care  of  the  reader. 

Returning  once  more  to  the  rigorous  equations  (65a)  we 
now  see  that  the  terms 


/  i  a  \ 

\fi-r 


represent  in  general  the  momentum  and  energy  (or  mass) 
acquired  by  'matter'  in  a  gravitational  field. 

The  four  equations  themselves  express  the  principles  of  momentum  and 
of  energy,  as  was  made  plain  above  on  their  appropriate  form.  I  avoid 
purposely  to  call  them  principles  of  'conservation'  of  momentum  and  of 
energy.  For  although  Einstein  succeeded  in  giving  them  the  form* 

*Sitzungsberichte  of  Berlin  Academy,  vol.  42,  1916,  p.  1115,  where  the 
German  T  and  /  are  the  above  V  —  g  Ttv—gt. 


ENERGY  PRINCIPLE  87 


in  which  they  would  -deserve  the  name  of  conservation,  yet  the  /^  built  up 
of  the  gM"  and  their  first  derivatives  has  not  the  character  of  a  general 
tensor,  but  behaves  so  only  with  respect  to  a  certain  class  of  coordinate 
systems  (for  which  g=  —  1).  In  view  of  this  it  has  not  seemed  necessary 
to  quote  here  the  values  of  the  t?  .  Suffice  it  to  say  that  since,  unlike  T*  f 
they  contain  only  the  gravitational  potentials  (and  their  first  derivatives), 
Einstein  calls  v  —  g  t*  'the  components  of  energy  of  the  gravitational  field', 
and  v  —  g  Ta  those  of  matter,  and  reads  the  last  set  of  equations:  the  total 
momentum  and  the  total  energy  of  matter  and  of  the  field  are  conserved. 
The  point  under  consideration  is  after  all  but  a  formal  one,  and  we  prefer 
therefore  to  content  ourselves  with  the  original  equations  (65a),  interpreting 
their  second  terms  as  momentum  and  energy  gained  (or  lost)  without 
attempting  o  locate  them  as  such  in  the  gravitational  field  before  their 
passage  to,  or  rather  appearance  in,  matter. 

Historically,  the  position  is  this.  In  the  special  or  restricted  theory  of 
relativity  the  principles  of  conservation  of  momentum  and  of  energy  were 
expressed  by  the  vanishing  of  the  'lor'  or,  in  Laue's  nomenclature,  of  the 
Divergence  of  a  world  tensor,  this  '  Divergence '  being  a  four- vector  whose 
components  were  transformed  by  the  Lorentz  transformation,  the  four 
equations  themselves  being  thus  invariant  with  respect  to  this  kind  of 
transformation.  The  tendency  to  imitate  these  principles  of  conservation 
in  the  generalized  theory  was  but  a  most  natural  one.  But  the  proper 
generalisation  of  that  special  Divergence  in  a  theory  admitting  any  trans- 
formations of  the  coordinates  is  the  Divergence  defined  in  the  general 
tensor  calculus,  i.e.,  the  contracted  covariant  derivative 

r,=  l?. 

of  the  mixed  tensor  of  matter  7** .  This  is  a  genuine  four- vector,  a  covariant 

tensor  of  rank  one,  and  the  original  generally  covariant  equations  (65), 

r«  =  o, 

are  the  only  appropriate  expression  of  the  principles  of  momentum  and 
energy.  Their  expanded  form  is  (65a)  and  this  cannot  in  general  be  given 
the  form  of  'conservation  laws'.  Only  for  constant  g«  ,  that  is  in  a  galilean 
domain,  does  it  reduce  to 

r~  T?  =  ° 

dxa 

which  is  identical  with  the  vanishing  of  what  was  called  the  Divergence  of 
ra  in  the  restricted  relativity  theory.  All  attempts  to  squeeze  the  broader 
Divergence  T*a  into  the  narrower  one  seem  artificial  and  useless.  For 
conservation  as  an  integral  law,  cf.  Einstein,  Berlin  Sitzungsber.,  1918, 
p.  448. 


88  RELATIVITY  AND  GRAVITATION 

29.  That  the  gravitational  field  equations  (together  with 
the  equations  of  motion  and  those  of  the  electromagnetic  field) 
can  be  deduced  from  a  single  variational  principle  or,  as  it  is 
called,  a  Hamiltonian  Principle,  was  first  shown  by  H.  A. 
Lorentz  (Amsterdam  Academy  publication  for  1915-16)  and 
by  D.  Hilbert  (Gottinger  Nachrichten,  1915,  No.  3),  and  later 
on  by  Einstein  himself.*  More  recently  Hilbert,  Weyl  and 
others  have  returned  to  this  subject  in  a  large  number  of 
publications,  in  some  of  which  the  importance  of  the  Hamil- 
tonian principle  seems  to  be  unduly  overestimated. 

Since  this  matter  is,  after  all,  of  a  purely  formal  nature, 
it  will  be  enough  to  give  here  but  a  very  brief  account  of 
Einstein's  own  treatment  as  developed  in  the  paper  just 
quoted. 

With  dx  as  a  short  symbol  for  dx\  dx2  dxz  dx±  Einstein 
writes  the  Hamiltonian  principle 

^~g(G+M)dx  =  0,  (68) 

where  G,  M  are  invariants.  Since  v  —  gdx,  the  volume  of  a 
world-element,  is  invariant,  so  also  is  the  whole  integrand. 
Let  M  be  a  function  of  the  g^  and  of  q  ,and  dqjdxa,  where  q, 
are  some  space-time  functions  describing  'matter',  while  G 
is  assumed  to  be  linear  in  d2g^  /  dxa  dx$  with  coefficients  de- 
pending only  upon  the  g^.  Then,  by  partial  integration, 


\ 


where  da  is  an  element   of  the  boundary  of  the  world-domain 
\dx  (the  particular  value  of  the  integrand  F  being  irrelevant), 

and  G*  is  a  function  of  the  g*1"  and  their  first  derivatives  only. 
Let  it  be  required  that  the  values  of  f  and  of  their  first 

derivatives  should  be  fixed  at  the  boundary.    Then  d    Fdcr  =  0, 

and  we  can  write,  instead  of  (68), 

*A  Einstein,  Hamiltonsches  Prinzip  und  allgemeine  Relativitdtstheorie, 
Sitzungsberichte  der  Akad.  der  Weiss.,  Berlin,  vol.  XLII,  1916,  p.  1111- 
1116. 


HAMILTONIAN  PRINCIPLE  89 

=  0,  (68a) 

where  the  whole  integrand  depends  upon  the  f  and  qt  and 
their  first  derivatives  only.  Thus  the  variation  of  the  g*v  gives 
at  once  the  ten  equations 


,          (69) 

dp 


where 
and 


Now,  let  G  be  the  curvature  invariant,  i.e.,  in  our  previous 
notation, 


Then,  on  performing  the  said  partial  integration,  it  will 
be  found  that 


[{7}  {?}-{:} 


With  this  value  of  H*  //?e  equations  (69)  become  identical 
with  Einstein1  s  field  equations  as  given  above,  if  we  put 


.e., 


or,  in  terms  of  the  covariant  tensor  of  matter, 

dM 


The  verification  of  this  statement  may  be  left  to  the  care  of 
the  reader  who  may  confine  himself  to   systems  for  which 


90  RELATIVITY  AND  GRAVITATION 

30.  Gravitational  waves.  Let  us  close  this  chapter  by 
briefly  mentioning  a  method  of  approximate  integration  of 
the  field-equations  given  by  Einstein  (Berlin  Academy  pro- 
ceedings for  1916,  p.  688)  which  exhibits  the  propagation  of 
gravitational  disturbances. 

Let  the  glK  differ  but  little  from  the  galilean  values,  in  a 
cartesian  system,  say,  or — in  our  previous  notation — let 

where  the  ylK  are  small.    Then  Einstein's  approximate  solution 
of  his  field-equations  is 

%. -T',, -KT'XX,  (72a) 

where  yfM  is  the  retarded  potential  of  —  2/c  TM  ,  that  is  to  say,  the 
familiar  particular  solution 

-  TIK  (x,  y,z,ct- r)dx dy  dz  (726) 

of  '  the  wave-equation ' 

'    =-2*7:,.  me) 


In  (72b)  r  is  the  three-dimensional  distance  of  the  point  for 
which  7'l/c  is  required  (for  the  instant  f)  from  the  integration 
element  dxdydz  at  which  the  value  of  TIK  prevailing  at  the 

instant  t  —  —is  to  be  taken. 
c 

This  solution  represents  gravitation  as  being  propagated 
with  the  normal  light  velocity  c,  the  slight  changes  of  the  latter 
due  to  the  gravitational  field  itself  being  manifestly  neglected. 
In  this  approximation  the  rigorously  non-linear  field  equations 
are  replaced  by  linear  differential  equations  of  the  form  (72c), 
the  usual  wave-equations. 

In  the  sub-case  of  a  stationary  gravitational  field,  when 
the  whole  tensor  of  matter  is  reduced  to  r44  =  p,  we  have  by 
(72b),  as  the.  only  surviving  y\K , 

,  K         \pdxdydz  x!2 

744=~2;j  \—r     '  2T- 


APPROXIMATE  INTEGRATION  91 

where  12  is  the  ordinary  newtonian  potential  of  the  gravitating 
masses,  and,  by  (72a),  the  only  surviving  yw  , 

*ft  2fi 

711  =  722=733=  —744=  — =     — -   , 

47T  C2 

so  that,  as  before,  the  role  of  the  potential  0,  is  taken  over  in 
part  by  —  %c*yu. 


CHAPTER  V. 

Radially  Symmetric  Field.   Perihelion  Motion,  Bending 
of  Rays,  and  Spectrum  Shift. 


31.  In  order  to  represent  the  motion  around  the  sun  of  a 
planet  as  a  'free  particle',  of  mass  negligible  compared  to 
that  of  the  central  body,  it  is  enough  to  find  a  radially  sym- 
metrical solution  of  Einstein's  field  equations  outside  the  sun, 

GIK  =  0,  (III0) 

considering  the  origin  r  =  0  of  polar  coordinates  r,  <£,  6  as  a 
singular  point. 

As  a  form  of  the  line-element,  sufficiently  general  for  this 
purpose,  let  us  assume 

ds*±  g!  dr*~-  r*atf  -  r2  sin20  dd*+gt  c*dt\  (73) 

where  gi,  g±,  written  instead  of  gu,  g44,  are  functions  of  r  alone, 
of  which  we  shall  thus  far  assume  only  that 

&(«>)  =  -1,     g4(«)  =  l,  (74) 

i.e.,  that  at  distances  r  large  compared  with  a  certain  length 
belonging  to  the  sun  (which  will  appear  in  the  sequel)  the 
line-element  tends  to  its  galilean  form  ds2=  —  dr2—  rz(d<j>2-\- 


Let  us  correlate  the  indices  of  the  coordinates  by  putting 

*i,  x2,  x3,  x4  =  r,  0,  0,  ct 

respectively.    Then  the  metrical  tensor  in  question  will  con- 
sist of  the  components 

2i  =  2iW,  &=  -r2,  £3=  -r2  sin20,  gi  =  g*(r),  (73a) 

where  gK  has  been  written  for  gKK  . 
In  the  more  general  case 

(j~1  _  „    x/Y  2 
«**    —  SK  UX'K    ) 

in  which  the  gK  =  gKK  are  any  functions  of  all  the  variables,  we 
have,  for  the  only  surviving  associated  tensor  components, 

92 


RADIALLY  SYMMETRIC  FIELD 


93 


f-  - , 

and  therefore,  recalling  the  definition  of  the  Christoffel  symbols, 


=  ,  for  all  i,  K  , 

«   »  2gK   dxt 


>-L    ,  for 


(75) 


while  all  other  Christoffel  symbols  vanish. 

Applying  these  formulae  to  the  more  special  tensor  (73o)  , 
writing 


and  using  dashes  for  derivatives  with  respect  to  r  =  #i,  we 
have  the  rigorous  values  of  the  only  surviving  Christoffel 
symbols,  altogether  nine  in  number, 


22 


23 


These  values  substituted  into  the  general  expressions  (55) 
for  the  components  GtK  of  the  curvature  tensor  give  zeros  for 
all  those  having  t^/c,  while  the  remaining  four  diagonal  com- 
ponents are,  rigorously, 


J_ 

6^22 


0 


(77) 


94  RELATIVITY  AND  GRAVITATION 

Thus  we  have,  according  to  the  field  equations  for  r>0, 
that  is,  outside  of  matter,  for  the  two  unknown  functions  gi, 
g2  the  three  differential  equations 

9/7    ' 

(a)     V+JJuW-V)-  —  =0 

r 


(b)  (/*/  -&/)+£!+  1=0 

(c)      V+V  =  0. 

The  last  of  these  equations  gives  hi+  h*  =  log  (gigO  =  const., 
that  is  to  say,  by  (74), 

gi  g4  =  const.  =  —  1  . 
Equation  (6)  now  becomes  g4+rg4'  =  l  or 


so  that  r(g4—  1)  =  const.  =  —  2L,  say. 

Thus  the  rigorous,  and  the  most  general,  solution  of  the 
field  Aquations  (6),  (c)  is  ultimately, 


where  L  is  any  constant,  soijie  length,  characterising  the  sun, 
i.e.,  here  the  singular  point  or  centre  of  the  gravitation  field. 
As  to  the  first  field  equation  (a)  it  is  satisfied  identically  by 
these  values  of  gi,  g4.* 

To  express  the  constant  L  in  terms  of  M,  the  sun's  mass 
in  astronomical  units,  we  may  apply  the  following  reasoning: 
As  we  already  know,  in  the  approximate  equations  the 


newtonian  potential  ft  =  M/r  is  replaced  by  --  744.       Now, 

2 

*In  fact,  since  gig*=  —1,  the  left  hand  member  of  equation  (a)  becomes 

k"+fc*4W/r, 

and  this  is,  by  (78), 

-~^  [4l-2r+2fr-2L)], 
which  vanishes  identically  for  all  r=*=2L. 


RADIALLY  SYMMETRIC  FIELD  95 

in  the  present  case,  744  =  ^4—  1=  —  2L/r.  Thus  M=c2L, 
whence 

.  •  ,  :  ,-,•;  £=£•         '..         (79) 

Ultimately  therefore,  the  line-element  (73)  corresponding 
to  a  radially  symmetrical  field  becomes,  rigorously, 

—  V<fc*-  (l  ~  —  )^2-rW+sinW02),       (80) 

a  form  of  the  solution  of  the  field  equations  first  given  by 
Schwarzschild  (Berlin  Academy  proceedings,  1916,  p.  189). 
As  was  already  mentioned  in  Chapter  IV,  the  dimensions  of 
L  as  defined  by  (79)  are  those  of  a  length.  This  length,  which 
is  sometimes  called  the  gravitation  radius  of  the  given  body, 
amounts  for  our  sun  to  about  1'47  kilometers.  Thus,  for  all 
applications  of  any  actual  interest,  L/r  is  a  small  fraction  and 
the  coefficient  of  dr2  can  be  replaced  by  —  (l  +  2L/r). 

32.  Perihelion  motion.  Let  us  now  consider  the  motion  of 
a  free  particle  (planet)  in  the  field  determined  by  the  line- 
element  (80),  that  is  to  say,  by  the  metrical  tensor 


(2L\  ~  1  1 

1-       -  ),  g2=~r2,  g3=-r2sin24>,  g4=  ---  . 
r  /  gl 


The  general  equations  of  motion  (15)  with  the  values  (76)  of 
the  Christoffel  symbols  become,  for  t  =  2,  3,  4,* 

d2<j>  2    dr    d(f> 

ds2  r    ds    ds 


E2     dr  d<f>  ~|  dO 

~r     <&          '       dTJds 


4          ,  ,   dr    dx* 
z  4    ^s    ds 


*Instead  of  the  first  equation  of  motion  (t  =  1)  it  will  be  more  convenient 
to  take  the  identical  equation  glK  Xt  XK  =1. 


96  RELATIVITY  AND  GRAVITATION 

Lay  the  plane  <£  =  7r/2,  the  equatorial  plane  of  the  coordi- 
nate system,  through  the  direction  of  motion  of  the  planet  at 
some  instant  t0.  Then,  at  that  instant,  d<f>/ds  =  Q  and  sin  20 
=  0,  and  therefore,  by  the  first  equation,  permanently  <£  =  ?r/2, 
that  is  to  say,  the  planet  will  describe  a  plane  orbit,  and  the 
remaining  two  equations,  together  with  the  identical  equation 

i  XK  =  1>  will  become 

0  +*L  0  =  o 


---V-ra#  =  l,  (80o) 

#4 


where  7z4  =  log  g4  and  g4=l  —  2L/r.     The  first  two  of  these 
equations  can  be  written 

4-log(r2l?)=0,    ™logfe4*4)=0, 
as  as 

and    give 

r*'$  =  p,  g&  =  k,  (81) 

where  p  and  k  are  arbitrary  constants.*    With  the  values  of 
#4  and  0  derived  from  (81)  equation  (80a)  becomes 

^ 
r* 

or,  putting  p  =  —  , 
r 

IA         1  O7" 

(82) 


P2        P2 

The  determination  of  the  orbit  is  thus  reduced  to  a  quad- 
rature. As  an  alternative  we  may  write  down  the  differential 
equation  of  the  orbit,  by  differentiating  the  last  equation  with 
respect  to  6, 

*The  first  of  (81)  represents  the  slightly  modified  law  of  Kepler:  areas 
swept  out  by  the  radius  vector  in  equal  proper  times  of  the  particle  (s/c 
instead  of  t)  are  equal. 


PERIHELION  MOTION  97 


Either  equation  differs  from  the  familiar  equations  of  celestial 
mechanics,  based  on  Newton's  principles,  only  by  the  under- 
lined last  term  of  the  right-hand  member. 

It  is  well  known  that  in  the  absence  of  this  supplementary 
term  the  orbit  is  a  conic  (an  ellipse,  a  parabola  or  a  hyperbola) 

p=  L_  [i+;€COs(0-«)]  (83) 

with  fixed  perihelion,  w  =  const.  In  fact,  equation  (82a)  is 
identically  satisfied  by  (83)  ;  and  so  is  (82)  if  we  put 

fh  (83o) 


so  that  the  orbit  is  an  ellipse,  a  parabola  or  a  hyperbola 
according  as  k2  is  smaller,  equal  to  or  greater  than  1  . 

In  general,  for  orbital  velocities  comparable  with  the  light 
velocity,  equation  (82)  gives  0  as  an  elliptic  integral  of  p,  to 
which  corresponds  a  complicated  non-closed  orbit.  Its  dis- 
cussion may  be  left  to  the  care  of  the  reader.*  Here  it  will  be 
enough  to  consider  small  velocities  such  as  occur  among  the 
planets  of  the  solar  system.  The  supplementary  term  is  then 
small  compared  with  the  newtonian  ones,  and  the  problem 
can  be  solved  approximately  by  a  conic  (83)  with  slowly 
moving  perihelion.  If  dp/  68  is  the  derivative  of  p  when  d>  is 
kept  fixed,  and  if  the  term  with  (d&/d#)*  is  neglected,  we  have 


V  =  (dJ-  +  *£ 
/     "  \d0         do> 


de  /      \ee   '  d&  de  J  :  \d0  /       de  d&  de  ' 

and  since  (dp/36)2  itself  accounts  for  the  first  three  terms  of 
the  right  hand  member  of  (82),  the  perihelion  motion  will  be 
determined  byf 

*Cf.  A.  R.  Forsyth,  Proc.  Roy.  Soc.,  XCVII  (1920),  p.  145,  also  W.  B. 
Morton,  Phil.  Mag.,  XLII  (1921),  p.  511. 

fThis  reasoning,  aiming  at  the  secular  motion  of  the  perihelion,  pre- 
supposes the  knowledge  of  absence  of  a  secular  variation  of  the  eccentricity 
€.  Cf.  footnote  on  p.  99,  infra. 


98  RELATIVITY  AND  GRAVITATION 

dp    dp   dw       , 
---  =  JL,p  . 

dO   d£  de 

Here  (83)  can  be  used  with  sufficient  accuracy  for  p  and  its 
two  derivatives,  so  that 

du  L          1  +  3e  cos  u  +  3e2cos2 


pk  snw 

where  w  =  0  —  co.  Integrating  this  from  0  to  2?r  over  0  or,  what 
for  our  approximation  is  the  same  thing,  over  u,  we  shall  have 
the  secular  perturbation  5o>,  the  motion  of  the  perihelion  per 
period  of  revolution.  The  second  and  the  third  terms  of  the 
integrand,  having  in  the  second  and  the  third  quadrants 
values  opposite  to  those  in  the  first  and  fourth,  contribute 
nothing  to  the  secular  perihelion  motion,  and  the  same  is 
true  of  the  first  term,  since  this  is  the  derivative  of  the  periodic 
function  —  cot  u.  We  are  thus  left  with 

27T 


=  _^L  f 

P*  \ 


5co  =  —  cot2«  dut 

P2  J 

o 

and  since  — cot2w  is  the  derivative  of  cot  u+u, 


*~  /5M. 

—  • 

This  being  essentially  positive,  the  secular  motion  of  the 
perihelion  is  progressive,  that  is,  in  the  sense  of  the  revolution 
of  the  planet. 

If  the  orbit  be  an  ellipse  (e2<l)  with  semi-axes  a,  b,  we 
have,  by  the  original  meaning  of  the  constant  p, 


= 


ds~       d  cT 

where  T  is  the  period  of  revolution,  and  by  (83), 


expressing  Kepler's  third  law.     Thus 


PERIHELION  MOTION  99 

Ju        27TQ2 


P 

and  (84)  becomes 

*2  ,0.   v 

(84a) 


-  e2) 

which  is  Einstein's  formula  for  the  secular  motion  of  the 
perihelion  of  a  planet,  undisturbed  by  other  planets,  per  period 
of  revolution.*  This  formula  gives  for  Mercury,  per  century, 
43"  or  43"  -1,  coinciding  most  remarkably  with  the  famous 
excess  of  perihelion  motion  of  that  planet,  unaccounted  for 
by  the  perturbations  due  to  the  other  members  of  the  solar 
family  of  celestial  bodies.  Although  the  rival  explanation 
based  on  perturbing  zodiacal  matter,  due  to  Seeliger  —  New- 
comb  (taken  up  more  recently  by  Harold  Jeffreys),  cannot 
be  considered  as  ultimately  discarded,  this  is  certainly  a 
most  conspicuous  achievement,  perhaps  the  greatest  triumph 
of  Einstein's  theory,  yielding  the  required  excess  without  the 
aid  of  any  new  empirical  constant  in  addition  to  the  light 
velocity  and  the  gravitation  constant.  As  to  the  remaining 
planets,  Einstein's  formula  gives  for  them  secular  perihelion 
motions  too  small  to  be  either  contradicted  or  confirmed  by 
observation  in  the  present  state  of  the  astronomer's  know- 
ledge. In  fact,  the  only  other  serious  anomaly  unaccounted 
for  by  newtonian  celestial  mechanics  (unless  Seeliger's  theory 
is  accepted)  is  the  excessive  motion  of  the  nodes  of  Venus,  but 
with  this  Einstein's  theory  is  essentially  powerless  to  deal, 
since  it  yields,  for  a  radially  symmetric  centre  of  course, 
rigorously  plane  orbits.  But  even  the  outstanding  node 
motion  of  Venus  is  generally  felt  to  be  much  less  important 
than  Mercury's  perihelion  motion  yielded  so  naturally  by 
Einstein's  theory  of  gravitation. 

*A  more  thorough  analysis  shows  that  this  is  the  only  secular  pertur- 
bation, the  eccentricity,  the  period  and  the  remaining  elements  of  planetary 
motion  being  unaffected  by  the  deviation  of  Einstein's  theory  from  that 
of  classical  celestial  mechanics.  Cf  .  W.  de  Sitter's  paper  in  Monthly  Notices 
of  the  Roy.  Astr.  Soc.,  London,  1916,  pp.  699  et  seq.t  more  especially  sec- 
tion 17. 


100  RELATIVITY  AND  GRAVITATION 

33.  Deflection  of  light  rays.  The  propagation  of 
light  is  given  by  the  minimal  lines  ds  —  Q  of  the  metrical 
manifold  determined  by  the  quadratic  form  (80)  .  By  reasons 
of  symmetry  it  is  again  sufficient  to  consider  the  plane 
0=  const.  =  7r/2.  Thus  the  light  equation  becomes 


£4  r 

If  v  be  the  system-velocity  or  the  'coordinate  velocity'  of 
light,  defined  by 


dt  /        \dt 
the  preceding  equation  gives 


and  if  77  be  the  angle  between  the  tangent  to  the  light  ray  and 
the  radius  vector,  so  that  dr/do-  =  cos  77,  rdd/d(r  =  sin  77, 

c2        1  Fees2??   .     .  9    ~1  ,OK, 

_  =  _       —  !  -fsm277      .  (85) 

V2          g4    L     £4  -I 

Thus,  if  the  ray  be  radial,  away  from  or  towards  the  origin, 
the  light  velocity  is  cg^  and  if  transversal,  c  Vjjfc  both  principal 
velocities  being  smaller  than  c,  and  both  tending  to  c  at 
infinity.  Neglecting  the  square  and  the  higher  powers  of 
L/r,  which  even  at  the  surface  of  the  sun  is  a  very  small 
fraction,  we  can  write,  approximately,  v/c  =F  1  —  (l+cos277)L/r. 
The  velocity  of  light  being  determined  by  (85),  the  shape 
of  the  ray  or  the  light  path  between  any  two  points  1,  2  can 
be  found*  by  means  of  Fermat's  principle 


!=5     -    =0. 
i  i 

In  fact  this  principle  can  be  proved    to   hold,  at   least   for 
stationary  gravitational  fields,  i.e.,  for  glK  not  containing  the 

*In  terms  of  r,  77,  and  thence  by  integration  in  terms  of  r ,  9. 


DEFLECTION  OF  RAYS  101 

time*  as  in  the  case  in  hand.  Those  interested  in  such  an 
application  of  Fermat's  principle  may  consult  de  Sitter's 
paper  quoted  in  the  preceding  section. 

But  a  much  more  speedy  way  of  obtaining  the  light  path 
is  to  consider  it  as  the  limiting  case  of  the  orbit  of  a  free 
particle.  In  fact,  returning  to  the  differential  equation  (82a) 
of  such  an  orbit,  and  remembering  the  original  meaning  of 
the  integration  constant  p, 

'•  ^T'       P         '          ' 

ds 

we  have  for  light,  or  for  a  'particle'  which  would  everywhere 
keep  pace  with  it, 

p=  oo, 

so  that  the  differential  equation  of  the  light  path  becomes 

||j  +p-3Z,P!  =  0.  (86) 

In  the  absence  of  the  last  term,  which  bears  to  p  the  small  ratio 
3L/r,  we  should  have  p  =  po  cos  6,  a  straight  line  whose  shortest 
distance  from  the  origin  is  TQ  =  1/po,  the  angle  6  being  measured 
from  the  corresponding  radius  vector.  Thus,  replacing  p  in 
the  last  term  by  po  cos  6,  which  amounts  to  neglecting  L*f- 
and  higher  order  terms,  we  have  for  the  light  ray  p  =  po  [cos  0-f- 
Lpo(l-fsin20)]  or 

—  =cos0+  — 
r  rQ 

The  angle  between  the  asymptotes  (r/rQ  —  oo  )  of  this  curve 
is  easily  found  to  be 


(87) 

TO         c0 

This  is  then  the  total  angle  of  deflection  of  a  light  ray  arriving 

*For  a  simple  proof  see  T.  Levi-Civita's  paper  in  Nuovo  Cimento,  vol. 
XVI,  1918,  p.  105.  Levi-Civita  assumes  also  gi4=g24=gs4  =  0.  The  latter 
limitation,  however,  does  not  seem  to  be  necessary.  Thus,  for  instance, 
it  can  be  shown  that  Fermat's  principle  leads  to  a  correct  result  in  the 
case  of  a  uniformly  rotating  system,  i.e.,  obtained  from  a  galilean  system 
by  the  transformation  6f  =  0+&tt  o>  =  const. 


102  RELATIVITY  AND  GRAVITATION 

from  a  distant  source  (star)  to  the  earth,  if  r0  be,  approxi- 
mately, the  shortest  distance  of  the  ray  from  the  origin, 
e.g.,  from  the  sun's  centre.  In  the  latter  case,  if  R  be  the 
sun's  radius,  we  have  4L/^  =  5'88/6'97.105  radians  =  1"75,  so 
that 

A  =  l"75-  , 

n> 

This  is  Einstein's  famous  formula  for  the  displacement  of 
star  images  seen  in  comparative  angular  proximity  to  the 
sun's  disc.  It  can  be  considered  as  fairly  well  verified  by  the 
results  of  the  Eclipse  Expedition  at  Sobral,  Brazil,*  of  May  29, 
1919,  which  were  ultimately  estimated  to  give,  when  reduced 
to  r0  =  R,  the  value  1"'98  with  a  probable  error  of  about  six 
per  cent.  This  is  certainly  more  than  a  mere  order-of-magni- 
tude  coincidence,  and  speaks  strongly  in  favour  of  Einstein's 
theory. 

The  displacements  according  to  Einstein's  formula  should,  of  course, 
be  away  from  the  sun  and  purely  radial.  The  displacements  measured 
on  the  Sobral  plates  deviated  from  radial  directions,  at  least  for  four  out 
of  the  seven  stars,  considerably,  to  wit  by  35°,  16°,  8°,  and  6°  for  the  stars 
numbered  11,  6,  2,  and  10,  whose  distances  from  the  sun's  centre  were 
about  8,  4,  2,  and  5R  respectively.  These  deviations  or  the  presence  of 
transversal  displacement  components  may  well  be  due  to  the  distortion 
of  the  coelostat-mirror  by  the  sun's  heat,  as  pointed  out  by  Prof.  H.  N. 
Russell.  Yet  a  refined  investigation  of  this  point  during  the  next  eclipse 
seems  very  desirable,  and,  as  I  understand,  will  be  taken  special  care  of 
at  the  Eclipse  Expedition  of  September  20,  1922,  at  which  it  is  designed 
to  avoid  the  use  of  a  mirror.  The  field  of  stars  near  the  sun,  during  totality, 
will  then  be  almost  as  favourable  as  in  1919.f 

34.  Shift  of  spectrum  lines.  Consider  an  atom,  say  of 
nitrogen,  placed  in  the  photosphere  of  the  sun,  at  rest  or 
practically  so.  Then  its  line-element  or  the  element  of  its 
'proper  time'  will  be,  by  (80),  and  writing  for  the  present  5 
instead  of  s/c, 


*The  measurements  of  the  Principe  Expedition,  made  under  un- 
favourable weather  conditions,  seem  by  far  less  reliable. 

fSome  preliminary  details  will  be  found  in  Monthly  Notices  of  the  Roy. 
Astr.  Soc.  for  May  1920,  p.  628. 


SPECTRUM  SHIFT  103 

and  any  finite  interval  of  its  proper  time 


R   /  \        R 

Let  another  nitrogen  atom  be  placed  in  one  of  our  terrestrial 
laboratories,  at  a  distance  r  from  the  sun's  centre.  Then  its 
proper-time  interval  will  be 


In  particular,  let  A£i  be  the  terrestrial,  and  At  the  solar  time 
period  of  one  of  the  natural  vibrations  or  spectrum  lines  of 
nitrogen. 

Now,  encouraged  by  the  traditional  belief  in  the  somewhat 
vague  'sameness'  of  atoms  of  a  given  kind,  Einstein  assumes, 
as  he  did  already  in  other  circumstances  in  the  special  rela- 
tivity theory,  that  the  said  two  atoms  are  'equal'  to  each 
other  in  the  sense  of  the  word  that  the  proper  times*  of  their 
vibration  periods  are  equal  to  each  other.  Eddington  in  his 
Report  (p.  56)  simply  says  that  an  atom  is  "a  natural  clock 
which  ought  to  give  an  invariant  measure  of  an  interval  ds, 
i.e.,  the  interval  ds  corresponding  to  one  vibration  of  the  atom 
is  always  the  same".  Weyl  states  the  case  in  an  apparently 
more  profound  way  by  saying  that  if  the  two  atoms  are 
"objectively  equal  to  each  other,  the  process  by  which  they 
emit  waves  of  a  spectrum  line,  when  measured  by  the  proper 
time,  must  have  in  both  the  same  frequency". 

In  short,  the  founder  of  the  theory,  as  well  as  his  exponents 
assume,  more  or  less  implicitly,  that 

As  =  A,yi. 

If  so,  then  the  ratio  of  the  solar  to  the  terrestrial  period  of 
vibrations  is 


r  /      \       R 

or,  since  in  our  case  R/r  is  but  a  small  fraction, 

—    =1+-   =l+2-109.1(T6.  (88) 

A/i  R 

*It  is  now  usual  to  extend  this  name  for  ds/c  from  special  to  general 
relativity  theory. 


104  RELATIVITY  AND  GRAVITATION 

Einstein's  conclusion  then  is  that  the  lines  of  the  solar 
spectrum,  compared  with  those  of  a  terrestrial  one,  should  be 
shifted  towards  the  red,  the  proportionate  increment  of  wave- 
length being 

5^   =  L  =2-109.10-°, 
X         R 

or  equivalent  to  a  Doppler  effect  due  to  a  (receding)  source 
velocity  of  0*633  kilometers  per  second.  This  amounts,  for 
violet  light,  to  about  O'OOS  A.  Now,  although  with  the 
modern  means  one-thousandth  of  an  A  or  even  less  can  be 
well  detected  in  comparing  spectra,  Dr.  St.  John  of  the  Mount 
Wilson  Observatory,  who  observed  43  lines  of  nitrogen 
(cyanogen)  at  the  sun's  centre,  and  35  at  the  limb,  was  unable 
to  detect  any  trace  of  the  predicted  effect.  His  observations 
were  made  and  discussed  in  1917,  and  his  final  conclusion 
then  was  that  "there  is  no  evidence  of  a  displacement,  either  at 
the  centre  or  at  the  limb  of  the  sun,  of  the  order  O'OOS  A". 
Since  that  time,  however,  in  view  of  the  entanglement  of  the 
Einstein  effect  with  shifts  of  a  different  origin,  and  seeing 
that  the  results  of  other  astrophysicists  were  not  quite  so 
definite,  Dr.  St.  John  suspended  his  final  judgment  and  is 
now  taking  up  a  thorough  discussion  of  the  whole  material 
of  solar  spectrum  shifts  from  E.  L.  Jewell's  first  observations, 
made  about  1890,  up  to  the  present.  The  natural  impression 
now  is  that  it  would  be  premature  to  either  assert  or  deny  the 
existence  of  the  gravitational  spectrum  shift. 

Einstein  himself  has,  on  more  than  one  occasion,  expressed 
the  very  radical  opinion  that,  should  the  shift  be  absent,  the 
whole  theory  should  be  abandoned.  Yet,  in  view  of  the  hypo- 
thetical nature  of  the  sameness  of  atoms  in  the  explained 
sense  of  the  word,  such  an  attitude,  though  personally  in- 
telligible, is  by  no  means  necessary.  It  is  true  that  the  in- 
variability of  an  atomic  ^-period  of  vibration  in  a  gravitational 
field  can,  with  the  aid  of  the  equivalence  hypothesis,  be  re- 
duced to  its  invariability  while  the  atom  is  being  moved 
about, — a  property  of  atoms  as  'natural  clocks'  already 


NATURAL  CLOCKS  105 

utilised  in  special  relativity.*  Yet  we  do  not  know  whether 
the  atoms  actually  possess  even  the  latter  property.  Thus, 
Einstein's  intransigent  attitude  proves  only  the  strength  of 
his  belief  that  the  atoms  are  or  will  turn  out  to  be  such 
natural,  ideal  clocks.  But,  after  all,  this  is  only  a  guess. 
A  very  reasonable  one  to  be  sure ;  for  if  not  among  the  atoms, 
then  there  is  indeed  but  little  hope  to  find  such  clocks  among 
other  'mechanisms',  natural  or  artificial. 

At  any  rate,  a  final  astrophysical  verification  of  Einstein's 
spectrum-shift  formula,  supported  perhaps  by  repeated 
experiments  on  canal  rays,  would  be  an  achievement  of 
fundamental  importance.  Until  then  'the  natural  clock'  will 
remain  a  purely  abstract  concept. 

*It  is  this  theoretical  attribute  of  atoms  which  has  led  to  the  conclusion 
that  moving  hydrogen  atoms  (canal  rays)  will  emit,  in  transversal  direc- 
tions, waves  (1—  iP/c*)"^  times  longer  than  atoms  at  rest.  But  even  this 
shift  effect,  though  tried  experimentally,  does  not  seem  to  have  ever  been 
detected. 


CHAPTER  VI 
Electromagnetic  Equations 


35.  Maxwell's  equations  of  the  electromagnetic  field  in 
empty  space  supplemented  by  the  convection  current  pV,  or 
the  fundamental  equations  of  the  electron  theory  are,  in 
three-dimensional  vector  notation,  with  x±  =  ct, 

—  -f-curlE  =  0,  divM  =  0 


dE  ,  ..       v     j.    _ 

---  h  curl  M  =  p  —  ,  div  E  =  p. 

8x4  c 

They  contain,  apart  from  the  velocity  v  of  moving  charges, 
but  two  vectors  E,  M  which  may  be  provisionally  called  the 
electric  and  the  magnetic  forces.  As  is  well-known  from  the 
special  relativity  theory,  these  equations  retain  their  form  or 
are  co  variant  with  respect  to  the  Lorentz  transformation, 
i.e.,  in  passing  from  one  to  another  inertial  system.* 

They  are  not,  however,  generally  covariant,  and  thus  not 
appropriate  to  the  purposes  of  the  general  relativity  theory. 

What  is  covariant  with  respect  to  any  coordinate  trans- 
formations is  the  somewhat  broader  system  of  equations, 
containing  two  more  vectors  D  and  B  which  may  be  called 
the  electric  and  the  magnetic  polarizations,! 

r)"R 

—  +curlE  =  0,  divB  =  0,  (A) 

8x4 

-   —  +curlM  =  p~,  divD  =  p.  (B) 

_  6X4  C 

*Cf.  for  instance  my  Theory  of  Relativity,  1914,  Chap.  VIII,  and,  for  the 
historical  aspect  of  the  subject,  Chap.  III. 

fOr  the  electric  displacement  and  the  magnetic  induction  respectively. 

106 


ELECTROMAGNETIC  EQUATIONS  107 

In  a  galilean  domain  or  an  inertial  system  D  and  B  reduce  to 
E  and  M  respectively,  but  in  general,  in  a  gravitational  field 
or  a  non-inertial  system,  the  polarizations  differ  from  the 
forces,  being  some  linear  vector  functions  of  the  latter. 

The  general  covariance  of  these  two  groups  of  electro- 
magnetic equations  was  first  noticed  and  developed  by 
F.  Kottler  as  early  as  in  1912*  and  shortly  afterwards,  with 
due  acknowledgement,  incorporated  by  Einstein  into  the 
physical  part  of  his  general  theory  of  relativity. 

Let  FIK  be  an  antisymmetric  covariant  tensor  of  rank  two 
or  a  six-vector,  which  will  embody  in  itself  B  and  E,  and  thus 
may  be  called  the  magneto-electric  six-vector.  Then  the  group 
(A)  of  equations  can  be  replaced  by  the  equations 

(Ai) 


d#x  dx,  dxK 

which  are  generally  covariant  since  their  left  hand  members 
are,  by  (46),  Chap.  Ill,  the  components  of  a  general  tensor  of 
rank  three,  the  antisymmetric  expansion  of  the  six-vector  FIK  . 
To  compare  (Ai)  with  (A)  and  to  see  the  simplest  form  of 
the  correlation  between  B,  E  and  the  six  components  of 
FIK  use  cartesian  coordinates  or,  in  the  presence  of  a  gravita- 
tional field  (always-  'weak'),  quasi-cartesian  coordinates  and 
denote  by  1,  2,  3  the  rectangular  components  of  B,  E  along 
the  three  axes.  Then  the  group  (A)  of  equations  will  be 


where  'etc.'  means  two  more  equations  by  cyclic  permuta- 
tion of  the  suffixes  1,  2,  3  only.  On  the  other  hand,  writing 
out  (Ai)  and  remembering  that  FIK  =  —FKt  ,  we  have 


*Friedrich  Kottler,  Raumzeitlinien  der  Minkowski'  schen  Welt,  Sitzungs- 
berichte  Akad.  Wien,  vol.  121,  section  Ila,  pp.  1659-1759. 

-8 


108  RELATIVITY  AND  GRAVITATION 


2S  u  2*  _ 

-  "         -  —  u,  etc. 

8x4          dxz          dx3 


2s    ,         Si  z  _Q 

dXi  dxz  dxs 

and  these  four  equations  become  identical  with  those  just 
written  if  we  put 

FM,  F3i,  Fiz  —  Bi,  B2,  Bs 

FU,  F%4,  7*34  =£1,  £2,  ES 

respectively,  or  more  compactly,  if  i,  k  be  reserved  for  1,  2,  3 
only, 

Fik=-B-,  *i4  =  E.  (89a) 

This  then  is  the  required  correlation  for  the  case  in  hand. 
Non-cartesian  coordinates  will  be  dealt  with  in  the  sequel. 

Next,  let  FIK  be  the  supplement  of  F^  defined,  as  in  (34),  by 
F'-ffF*.  (90) 

Then  the  group  (B)  of  the  electromagnetic  equations  will  be 
replaced  by  the  four  equations 

™    -£-(Vg    FtK)  =  C\  (50 

Vg      dxK 

where  Cl  is  a  contravariant  four-vector.  Such  also  being  the 
left  hand  member,  the  divergence  of  FtK  ,  as  in  (47),  the  equa- 
tions (Bi)  will  be  generally  contravariant.  To  compare  them 
with  (B)  and  to  find  the  correlation  proceed  as  before.  Thus, 
on  the  one  hand, 


=  p—  ,  etc. 

dx2  dx3  c 


i  +    i   . 


- 


and  on  the  other  hand,  remembering  that  FKK  =  0  and  F1"  = 
-F", 


ELECTROMAGNETIC  EQUATIONS  109 


Vo,  etc., 
/ 


etc. 

dx\ 

The  required  correlation  is,  therefore, 

l  D 


or,  in  the  previous  abbreviated  notation, 

Since  FIK  is  thus  seen  to  embody  the  electric  polarization  and 
the  magnetic  force,  it  may  be  distinguished  from  its  supple- 
ment by  the  name  of  the  electro-magnetic  six-vector.  At  the 
same  time  we  have,  by  comparing  the  right-hand  members  of 
the  two  forms  of  equations, 

V  c       c       c 
or,  more  shortly, 

s  ~  N. 

(91) 


V- 

exhibiting  CK  as  the  electric  four-current.     It  is  interesting  to 
note  that  since  we  can    put  vi/c=dxi/dx4:  and 
the  last  correlation  can  also  be  written 


V  —  g    dx4 

Since  dxK  is  a  contravariant  vector  as  well  as  the  four-current,  the 
factor  of  dx  K  will  be  an  invariant,  and  since  V  —  gdxi  dx^dxz  dx4  is  also 
an  invariant,  the  volume  of  a  world-element,  we  see  that  the  electric  charge 
de=pdx\  dxzdxs  is  again  an  invariant.  Then,  however,  not  p  itself  but  p 
divided  by  the  determinant  —  \gik\  will  be  the  system-density  of  electricity. 


110  RELATIVITY  AND  GRAVITATION 

It  may  be  well  to  illustrate  the  general  transformation  formulae  of 


by  writing  them  out  for  the  simplest  case  of  two  inertial  systems  S,  S'  in 
uniform  translational  motion  relatively  to  each  other.  The  transformation 
is  in  this  case  the  familiar  Lorentz  transformation,  i.e.,  in  cartesian  co- 
ordinates and  with  the  Xi  axis  along  the  direction  of  motion, 


where  p=v/c  and  7  =  (1—  (&}  ~  #f  if  v  be  the  velocity  of  S'  relatively  to  S. 
First  of  all,  since  in  this  case  the  gtK  have  their  galilean  values  (in  both 
systems),  we  have 

B  =  M,  D=E, 

so  that  there  is  no  need  to  consider  the  supplement  of  FtK  ;  it  is  enough  to 
treat  FtK  itself.  Next,  since  x2,  x3  depend  only  on  x2t  x3',  being  equal  to 
them  respectively,  we  have 


Similarly, 


i.e., 

Af2' 

and  so  on.    Thus  we  get  the  transformation  formulae 


familiar  from  the  special  relativity  theory.     The  corresponding  transform- 
ation of  the  four-current  may  be  left  as  an  exercise  for  the  reader. 

It  will  be  kept  in  mind  that  the  correlations  of  the  forces, 
the  polarizations  and  the  current  and  charge  density  to  the 
two  conjugated  six-vectors  and  the  four-current  given  in 
(89a),  (896),  (91)  are  valid  only  for  the  particular  case  of  a 
cartesian  or  quasi-cartesian  coordinate  system.  With  other 
systems,  such  for  instance  as  the  polar  coordinates,  even  in  a 
galilean  domain,  the  correlation  formulae  are  more  compli- 
cated, and  contain  besides  the  determinant  g  the  several 
components  gllc  of  the  metrical  tensor  or  (in  a  non-galilean 
domain)  parts  of  them,  as  will  be  seen  later  on.  It  is  important 


ELECTROMAGNETIC  EQUATIONS  111 

to  understand  that  there  is  nothing  general  about  these 
correlations,  apart  from  the  fact  that  FIK  embodies  somehow 
the  three- vectors  B  and  E,  and  F^  the  vectors  D  and  M,  and 
C"  the  convection  current  and  the  charge  density,  everything 
being  entangled  with  the  metrical  tensor  and  through  it  also 
with  gravitation. 

From  the  standpoint  of  general  relativity  the  master 
equations  are  henceforth  no  more  the  broadened  maxwellian 
equations  (A),  (B)  but  the  set  of  generally  covariant  or 
contra  variant  equations  (Ai),  (Bi)  with  the  metrical  link  (90) 
between  the  two  six-vectors.  It  will  be  well  to  gather  here 
these  somewhat  scattered  equations;  the  whole  generally 
covariant  electromagnetic  set  is  thus 


dxt  dxK 

_L  A  (Vg>)= 

V       dxK 


(IV) 


This  will  read  as  follows:  the  expansion  of  the  magneto- 
electric  six-vector  vanishes;  the  divergence  of  the  electro- 
magnetic six-vector,  the  supplement  of  the  former,  is  equal  to 
the  electrical  four-current. 

36.  The  four-potential.     Manifestly,  the  first  of  the  equa- 
tions (IV)  will  be  identically  satisfied  if  we  put 

^  =  ^  _  a*^ 

dxK          dxt 

where  0t  is  a  covariant  vector.  If  this  be  substituted,  the 
six  terms  destroy  themselves  in  pairs,  and  the  covariant 
nature  of  0,  ensures  the  required  tensor  character  of  FtK ,  the 
rotation  of  0t  (cf.  p.  61).  The  latter,  which  is  seen  to  embody 
Maxwell's  vector  potential  and  the  electrostatic  potential,  is 
called  the  four-potential. 

With  the  correlation  (89a)  the  six  equations  (92)  become 


112  RELATIVITY  AND  GRAVITATION 


or 

B=curlA,      E=-  —  -V0, 
cot 

exhibiting  the  three-dimensional  vector  A=—  (0i,  02,  0s)  as  Maxwell's 
vector-potential  and  0  =  04  as  the  electrostatic  potential. 

The  first  group  of  equations  (IV)  being  thus  satisfied  by 
(92),  the  second  group  gives 

-*»)>€•,  (93) 

d*tt    AJ 

which,  assuming  gtK  to  be  known,  are  four  differential  equa- 
tioijs  o£  the  second  order  for  as  many  components  of  the  four- 
'  current.  Since  the  four-potential  enters  only  through  its 
rotation,  we  can  without  loss  to  generality  subject  its  com- 
ponents to  a  kind  of  solenoidal  condition,  as  follows.  If 
0*  =  g*B0o  De  the  associated  four-potential,  a  contravariant 
vector,  then  its  divergence  defined  by  (48)  is  a  general  in- 
variant or  scalar,  and  the  condition  in  question  can  be  written 

£-(Vg>)=0.  (94) 

In  a  galilean  domain  the  equations  (93),  (94)  become 

--  V2A=  —  Pv.    -^L-v24>  =  p 
c*dt\  c          c*dP 

div  A+  1   *±  =  0, 
c    dt 

the  familiar  equations  of  the  electron  theory  for  the  vector 
potential  A  =  —  (<£i,  <£2,  <fo)  and  the  electrostatic  potential 
0  =  04-  In  general,  however,  the  equations  (93)  for  the  four- 
potential  will  contain  in  a  complicated  way  the  components 
of  the  metrical  tensor,  which  again  means  an  entanglement 
of  the  electromagnetic  with  the  gravitational  field.  This 
mutual  relation  of  the  two  fields  appears  directly  in  the  third 
of  equations  (IV)  giving  the  general  connection  between  the 
magneto-electric  six-vector  and  its  supplement. 


THE  FOUR-  POTENTIAL  113 

Since  the  four-potential  is  a  covariant  and  dxK  a  contra- 
variant  vector,  their  inner  product 

dl  =  4>KdxK  (95) 

is  an  invariant.  This  invariant  linear  differential  form  plays 
the  same  role  with  respect  to  electromagnetism  as  the  quad- 
ratic differential  form 


with  respect  to  gravitation.  As  the  latter  determines,  inter 
alias,  the  gravitational  field,  so  does  the  former  determine  the 
electromagnetic  field.  This  is  only  a  different  way  of  stating 
that  the  <j>K,  the  coefficients  of  dl,  determine  the  electromag- 
netic, similarly  as  the  gtK  determine  the  gravitational  field 
together  with  the  riemannian  metrical  properties  of  space- 
time.  Recently  a  differential  geometry  somewhat  broader 
than  Riemann's  was  proposed  by  Weyl  who  goes  deep  into 
the  matter  and  attributes  to  the  linear  differential  form  an 
equally  fundamental  metrical  (gauging)  function  as  to  the 
quadratic  differential  form.  But  reasons  of  space  prevent  us 
from  entering  here  into  this  subject,  and  the  interested  reader 
must  be  referred  to  Weyl's  own  book*  for  further  information. 
Moreover,  these  new  physico-geometrical  speculations, 
although  undoubtedly  attractive,  are  still  being  debated 
between  Weyl  and  Einstein,  f  and  may  therefore  be  appro- 
priately omitted  in  a  book  of  the  present  type. 

37.  Let  us  once  more  return  to  the  electromagnetic  equa- 
tions (A),  (B)  in  order  to  compare  them  with  the  tensor 
equations  (IV)  for  the  case  of  a  non-cartesian  system  of  space 
coordinates.  As  a  good  example  of  this  kind  we  may  take 
any  orthogonal  curvilinear  coordinates  xi}  x2,  #3.  It  is  well 
known  that  if  the  space  line-element  in  these  coordinates  be 
given  by 


(96) 

wz  Ws  Wi 


*H.  Weyl,  Raum-Zeti-Materie,  3rd  ed.,  Berlin  1920,  §16  and  §34. 
fCf.  Einstein's  remarks  to  Weyl's  paper,  with  Weyl's  reply,  in  Berlin 
Sitzungsber.,  1918,  and  Einstein's  recent  paper,  ibidem,  1921,  pp.  261-264. 


114  RELATIVITY  AND  GRAVITATION 

and  if  Rt  be  the  components  of  a  three-vector  R  tangential 
to  the  Alines  of  the  network, 

(97) 


/  &  A  d    /  R2\  d    /  Ra  VI 

1    -  -     I    +    -  I  -         -    I     +     -  1     •—    I 

i  \W2W3/        dx2\w^Wi/         dXt\WiWt/-l 
and  the  curvilinear  components  of  curl  R  are 

(curl  H)t  -«*  J~-  (  -3  )   -   I/  *  yi  etc.  (98) 

Ld#2  \  w3  /         dx3\W2/-} 


With  these  expressions  the  group  (A)  of  equations  becomes, 
provided  of  course  that  the  wf  are  independent  of  time, 


dxi 


dx2  \W3  Wi 

and  similarly  for  the  group  (B)  of  electromagnetic  equations. 
These  equations  are  to  be  compared  with  the  first  and  second 
of  the  tensor  equations  (IV).  To  find  the  required  correlation 
in  terms  of  the  gtK  notice  that  if  the  domain  is  assumed  to  be  a 
galilean  one,*  we  have 

ds2  =  gllc  dxL  dxK  =  dx<i2  —  da2, 
so  that 

g-  =  ~—     *44=1 
* 

and  the  remaining  gM  vanish.  Under  these  circumstances 
the  comparison  gives  at  once,  with  gt  written  for  ga, 


*Otherwise,  say  in  the  presence  of  gravitation,  not  the  whole  of    —    is 

«i 

to  be  thrown  upon  the  coefficient  of  dxi  in  the  expression  for  the  length 
dffi  considered  from  the  system-point  of  view. 


ORTHOGONAL  COORDINATES  115 


etc.  ;   F4i  =  V-  ^  Eb  etc. 


etc. 


V— 


etc.,  C4  =  p, 


(99) 


which  is  the  required  correlation. 

The  relations  between  the  polarizations  and  the  forces, 
determined  in  general  by  the  third  of  equations  (IV),  follow 
easily.  In  fact,  since  in  the  present  case  gu  =  1/&,  |^*  =  1,  and 
the  remaining  glK  vanish,  we  have 


that  is  to  say, 


and  therefore,  by  (99),  Bi  —  Mi,  etc.,  and  Ei—Di,  etc.    In  fine, 
B  =  M,  D  =  E, 

the  polarizations  are  identical  with  the  forces,  and  the  equa- 
tions (A),  (B)  reduce  to  the  usual  electromagnetic  equations 
for  the  vacuum,  giving  c  as  light  velocity,  and  so  on.  This 
result  might  have  been  expected,  for  the  present  case  differs 
from  that  corresponding  to  ds^^dx^—dx^—dx^—dx^  solely 
by  the  use  of  curvilinear  instead  of  cartesian  coordinates. 

38.  Let  us  now  consider  the  relation  between  B,  D  and 
M,  E  in  another  example  which,  besides  being  instructive 
in  a  general  way,  will  show  how  the  propagation  of  electro- 
magnetic waves  is  influenced  by  gravitation. 

If  a  system  be  used  for  which  £41  =  £42  =  £43  =  0,  the  four- 
dimensional  line-element  can  be  written 

ds2  =  £44  dx?  +  gik  dxi  dxk  ,  ».  *  =  i  ,  2,  3.  (  1  00) 


In  a  weak  gravitational  field  £44  as  well  as  the  g,*will  differ 
but  little  from  their  galilean  values.     Thus,  if  the  #,-  are 


116  RELATIVITY  AND  GRAVITATION 

cartesian  or  quasi-cartesian  coordinates,  g44  and  the  gu  will 
differ  but  little  from  +1  and  —1  respectively,  and  the  remain- 
ing gik  will  be  small  fractions.  Thus,  from  the  system-point 
of  view,*  the  electromagnetic  equations  (A),  (B)  will  be 


so  that  a  comparison  with  the  tensor  equations  will  give  again 
F,*=B,  F,4  =  E;^=-^-M,    7*'=  -=-D  ,          (89) 

V-g  V-OT 

as  in  (89a),  (896). 

Since  g^  =  0,  the  general  relation  FiK  =  gla  gK$  F^  between 
the  two  six-  vectors  will  now  give 


and  two  similar  equations  for  F3i,  Fi2.    But  these  are  the  solu- 
tions of  the  three  equations 

F™=  —  (gn  ft$+giaFn-f  fijFaX  etc., 
h 

where  h  is  the  determinant  |  gik  \  .    Now,  h  =  g/gu,  and  therefore 

F*s  =  —  (gnF23+guF31+g13Flz),  etc. 

g 
Again, 

F*=g4*gifiF*  =  gugli  F*,  etc, 
i.e., 


and  two  similar  equations  for  F&,  F^.     Whence,  by  (89), 

£44 

M\  =  —     .  -  (gn-Bi  +  gi2#2  +  ^13-^3)  ,  etc  . 
V-g 

,  etc, 


V- 


g 


*Analogously  to  the  sense  in  which  'the  system- velocity '  of  light  was 
used  previously,  and  contrasted  with  the  local  point  of  view. 


LIGHT  IN  GRAVITATION  FIELD  117 

or,  solving  for  the  polarisation  components  and  noticing  that 


,  etc. 
,  etc. 

Thus  B  is  exactly  the  same  linear  vector  function  of  M 
as  is  D  of  E.  Introducing  the  symmetrical  linear  vector 
operator 


-co 


g11    £12 


32 


(101) 


we  can  write  shortly 


where 


(102) 
(103) 


In  absence  of  gravitation  the  glK  assume  their  galilean  values, 
the  operator  co  becomes  an  idemf actor,  g44=l,  and  /x=J£  =  l, 
giving  B  =  M  and  D  =  E.  From  the  system-point  of  view  the 
vacuum  is  thus  transformed  by  gravitation  into  a  crystalline 
electromagnetic  medium  with  anisotropic  permeability  n  and 
permittivity  K.  These  operators  have,  however,  by  (103), 
at  every  point  common  principal  axes  (which  are  orthogonal) 
and  the  same  principal  values.  Now,  owing  to  this  peculiarity 
the  velocity  of  propagation  of  an  electromagnetic  wave, 
although  varying  from  point  to  point  and  dependent  upon 
the  direction  of  the  wave-normal,  can  be  easily  proved  to  be 
independent  of  the  orientation  of  the  light  vector  D.  Thus, 
although  the  medium  is  anisotropic,  there  will  be  no  double 
refraction  due  to  the  gravitational  field.*  In  fact,  if  n  be  the 
wave-normal  and  t)  the  velocity,  that  is  the  system-velocity 


*Cf.  in  this  connection  A.  O.  Rankine  and  L.  Silberstein,  Propagation 
of  light  in  a  gravitational  field,  Phil.  Mag.,  vol.  39,  1920,  p.  586. 


118  RELATIVITY  AND  GRAVITATION 

of  propagation,  along  the  wave-normal,  we  have  from  the 
electromagnetic  equations  (^4),  (B),  (102),  wtth  p  =  0, 

-#E  =  FMn,  -MM=  FhE,  (104) 

0  0 

as  will  be  seen  at  once  by  considering  a  wave  of  discontinuity 
and  using  the  general  compatibility  conditions  given  else- 
where.* Now,  since  the  operator  K  is  identical  with  M>  the 
last  two  equations  give 

'-  ^E+Fn(^~1FnE)=0, 

Cf 

for  every  direction  of  E.  Here  the  operator  K  "  l  is  the  inverse 
of  K.  If  KI,  etc.,  be  the  principal  values  of  K  and  n\,  etc.,  the 
components  of  n  or  the  direction  cosines  of  the  wave-normal 
with  respect  to  the  principal  axes,  the  last  equation  gives  at 
once 


i.e.,  a  propagation  velocity  independent  of  the  orientation  of 
the  light  vector,  which  proves  the  statement. 

If  gi>  £2,  £3  are  the  principal  values  of  the  vector  operator 
gut  £i2>  •  •  •  £33,  the  inverse  of  -co,  then  the  principal  values  of 
-co  itself  are  1/gi,  etc.,  and  we  have,  by  (103)  and  since 


,inf- 
—     +    —    +    —      •  (105) 

gi  g2 

Such  being  the  formula  for  the  velocity  of  propagation  on 
the  electromagnetic  theory  of  light,  it  is  interesting  to  com- 
pare it  with  the  light  velocity  v  yielded  directly  by  Einstein's 
fundamental  equation  ds  =  Q.  This  velocity  is  taken  along 
'the  ray'  instead  of  the  wave-normal.  Thus,  by  (100),  if  u 
be  a  unit  vector  along  the  ray,  and  Ui  its  direction  cosines, 

*L.  Silberstein,  Annalen  der  Physik,  vol.  26,  1908,  p.  751  and  vol.  29, 
1909,  p.  523,  or  Theory  of  Relativity,  London,  1914,  p.  56. 


PONDEROMOTIVE   FORCE  119 

c-  dxi  dxk 

£44  —  =  —  gik   -  =  —  gikUiUkj 

v2  da    dor 

and  especially  if  HI  be  the  direction  cosines  with  respect  to  the 
principal  axes  of  the  operator  gn,  gi2,  .  .  .  £33, 

.  (106) 


V2  g44 

Formula  (85),  used  in  connection  with  the  bending  of  rays  around  the 
sun,  is  only  a  special  case  of  (106).  In  that  case  the  principal  axes  are 
along  the  radial  and  all  the  transversal  directions,  while  the  principal  values 


g44  £4 

and  Uiz  =  cos2r),  «22+«32  =  sin2r7,  so  that~(106)  reduces  to  (85). 

If  the  wave-normal  n  coincides  with  a  principal  axis,  say 
with  the  first  one,  we  have,  by  (105),  tf/cz  =  —gu/&j  and  by 
(106),  c2/vz=  —  gi/g44;  that  is  to  say,  v  =  b,  as  it  should  be.  For 
then  the  ray  falls  into  the  wave-normal.  But  in  general  the 
ray  does  not  coincide  with  the  wave-normal,  and  so  does  v 
differ  from  *)•  The  question  whether  the  null-line  equation 
(106)  is  always  compatible  with  the  electromagnetic  equation 
(105)  may  be  left  to  the  care  of  the  reader.  If  the  ray  be 
defined,  as  usual,  by  the  Poynting  flux  of  energy,  its  direction 
will  be  that  of  the  vector  product  FEM,  and  all  questions 
concerning  the  light  ray  will  follow  from  (104)  with  K  =  p  as 
given  by  (103). 

39.  Ponderomotive  force,  and  energy  tensor  of  the  electro- 
magnetic field.  The  general  tensors  corresponding  to  these 
were  easily  suggested  by  the  results  already  known  from  the 
special  relativity  theory. 

The  inner  product  of  the  magneto-electric  six-vector  and 
the  four-current,  i.e.,  the  covariant  vector 

PI-F«C;  (V) 

gives  the  ponderomotive  force  on  a  charge,  per  unit  volume, 
together  with  its  activity  or,  in  other  words,  the  momentum 
and  the  energy  transferred,  per  unit  volume  and  unit  time, 
from  the  electromagnetic  field  upon  the  electric  charges. 


120  RELATIVITY  AND  GRAVITATION 

In  fact,  using  for  instance  cartesian  coordinates  and  g  =  —  1, 
we  have  for  the  first  three  components  of  Pt,  by  (89)  and  (91), 

Pi  =  P  f—  (v*B3  -v3Bz)  +EiJ,  etc., 
or  if  PI,  P2,  Pa  be  condensed  into  the  three-vector  P, 


which  is  the  familiar  formula  for  the  ponderomotive  force, 
while  the  fourth  component  becomes 

p4=  -  -£  (EM+E^+EM)  =  -  -*-  (Ev) 

c  c 

or,  sinceVFvB  =  0, 


which,  apart  from  the  factor  —l/c,  is  the  activity  of  P. 
Somewhat  more  generally,  the  same  formulae  will  hold  with  p 
replaced  by  p/V  —  g  . 

But  it  will  be  understood  that  from  the  standpoint  of 
general  relativity  the  master  formula  for  the  electromag- 
netic momentum  and  energy  transfer  is  again  (V),  as  were 
before  the  electromagnetic  field  equations,  all  generally 
covariant. 

By  means  of  (IV)  and  (V)  the  four-force  Pt  can  be  repre- 
sented as  the  covariant  derivative  of  a  second  rank  tensor, 
a  generalization  of  the  array  of  maxwellian  stresses,  momen- 
tum and  energy  density.  Following  Einstein's  example  it  will 
be  enough  to  give  here  the  required  form  of  Pt  for  such  co- 
ordinates for  which  g=  —  1,  and  therefore,  by  the  second 
of  (IV), 


Thus,  by  (V), 


ENERGY  TENSOR  121 

The  second  term  is,  by  the  first  of  the  equations  (IV), 

K\     ,     ^^x 
t  dxK 


dx.  dxt  dxK 

But  the  bracketed  expression  vanishes.  In  fact,  since  the 
summation  is  to  be  extended  over  all  K,  X,  and  since  both  F- 
tensors  are  antisymmetric,  this  expression  can  be  written 


dxt  dxK  dy. 

to  be  summed  only  over  K  <  X.  But  the  third  term  of  the 
bracketed  factor  is  +dFlK  /d#x,  so  that  the  whole  factor  of 
F0  vanishes,  by  the  first  of  (IV).  Thus 

•j  E"  r)  7?  f)  p? 

<9v»  r)v  r)T 

O^X  OJii  oxt 

and  since  here  K,  X  can  be  replaced  by  a,  /3  and  vice  versa, 


The  last  term  can  be  transformed  into  —  J  FKr  FK^  gxp  dgpr  /dxt  , 
so  that 

P.  =  -^-  (PM  F*)-  }  —   (F.x  F^J-JPA/^^  ^-r  . 
d#x  5aft  drct 

Finally,  if  we  denote  by  F  the  invariant  FK^  7r<cX  and  in- 
troduce the  mixed  tensor 

T*  =iF£  -FMP*-,  (107) 

the  last  formula  becomes 


i«=   71,  (108) 

to.  d*. 


122  RELATIVITY  AND  GRAVITATION 

exhibiting  the  four-force  in  terms  of  77,  the  energy-tensor  of 
the  electromagnetic  field. 

To  recognize  in  the  latter  an  old  friend  consider  a  galilean  domain  and 
use  cartesian  coordinates.  Then,  the  glK  being  constant,  (108)  reduces  to 
the  familiar  equation 

P  _  «*: 

£-' 

and  since,  by  (89),  in  the  present  case,  F-&  =  F2Z  =  Mit  etc.,  Fu=—Fi4  = 
EI,  etc.,  we  have 

F 
and  (107)  gives 


etc., 

which  are  Maxwell's  electromagnetic  stress  components,*  further 
TV  =  T41  =  -  (E2MZ  -EzMz~),  etc., 

which  are  the  negative  components  of  the  energy  flux  divided  by  c,  or  the 
components  of  electromagnetic  momentum  per  unit  volume,  and  finally 


which  is  the  negatived  density  of  electromagnetic  energy. 

The  right  hand  member  of  (108)  can  be  shown  to  be  the 
divergence  of  the  mixed  tensor  27  or  its  contracted  covariant 
derivative  T?a  as  defined  by  (4J).  In  fact,  since  for  constant 


,  <          f  =  0, 


g,  by  (67)  ,  <  =  0,  the  said  divergence  reduces  to 


and  since  in  our  case  Tp  is  symmetrical,  this  can  be  shown 
to  be  identical  with 

•j  ya  r\    K\ 

27*=^    +i-  -TA-,  (42c) 

dxa  dxt 

where  TKX  =  gKV  T^.      On  the  other  hand,  since  g*^  gKV  is  a 
constant,  to  wit  5^,  we  have 


[a  relation  to  be  used  also  in  passing  from  (426)  to  (42c)],  and 
^Tensions  proper  being  counted  positive. 


ENERGY  TENSOR  123 

the  second  term  of  (108)  becomes  identical  with  the  second 
term  of  (42c).     Thus 

P.-7T.  =  Div(77)  (1080) 

exhibiting  the  four-force  as  the  divergence  of  the  mixed  energy- 
tensor  of  the  electromagnetic  field. 

If  the  electric  charges  are  under  the  exclusive  control  of  the 
electromagnetic  field,  the  total  four-force  Pt  vanishes,  and  we 
have 

IT.=Div  (77)  =0.  (109) 

These  four  equations  are  perfectly  analogous  to  the 
'equations  of  matter',  (65),  given  in  Chapetr  IV,  the  'tensor 
of  matter'  being  now  replaced  by  the  energy-tensor  of  the 
electromagnetic  field  defined  in  (107).  These  equations 
express  in  either  case  the  principles  of  energy  and  of  momen- 
tum. 

Instead  of  the  mixed  tensor  (107)  we  can  introduce  the 
co variant  electromagnetic  tensor  giVTyK  =  TlK.  If  the  form 
(III)  of  the  gravitational  field-equations  be  used,  then  in  the 
presence  of  an  electromagnetic  field  the  components  of  the 
latter  tensor  (multiplied  by  the  gravitation  constant)  have 
to  be  included  in  the  corresponding  components  of  the  tensor 
of  matter  appearing  in  the  right-hand  member  of  those 
equations.  Thus  both  kinds  of  stresses,  energy,  etc.,  con- 
tribute to  the  curvature  tensor  GIK  and  through  it  codetermine 
the  gravitational  field.  The  contributions  of  the  electro- 
magnetic tensor  components  are,  of  course,  for  all  technically 
obtainable  fields,  exceedingly  small  as  compared  with  those 
due  to  matter  in  the  narrower  sense  of  the  word.  Theoreti- 
cally, however,  the  r61es  of  the  two  kinds  of  energy-tensors 
are  equivalent. 


—9 


APPENDIX. 

Manifolds  of  Constant  Curvature. 


As  was  mentioned  in  Chapter  III,  an  ^-dimensional 
manifold  of  constant  isotropic  riemannian  curvature  K,  posi- 
tive, nil  or  negative,  is  characterized  by  the  differential 
equations  (54),  which  can  be  deduced  from  the  general 
formula  (53)  for  the  riemannian  curvature.*  If  we  put 


where  R  may  be  any  constant,  imaginary  or  real,  finite  or 
infinite,  the  said  equations  are 

(iX,  tuc)  =  —  (gtM  S\K  ~  £«  &*)  i  (HO) 

to  be  satisfied  for  all  t,  X,  K,  ju.  In  order  to  pass  from  Riemann's 
covariant  symbols  to  the  mixed  curvature  tensor  use  (50a). 
Thus,  multiplying  both  sides  of  (110)  by  gXa  and  taking 
account  of  (32), 


Einstein's  tensor  G«   is  the  contracted  curvature  tensor 


CB  --<«&.-«:&.)- 

Kf- 
The  first  term  in  the  brackets  is  simply  g^  ,  while  the  second, 


*Cf.  also  W.  Killing,  Die  Nicht-Euklidischen  Raumformen,  Leipzig  1885, 
Section  123. 

124 


CufcVAfORE 

in  which  5"  or  1  is  to  be  taken  n  times,  is  equal  ng^  .     Thus, 
for  a  manifold  of  n  dimensions,  of  the  said  kind, 

G«  -  -  ft.  ,  (111) 


for  all  values  of  i,  K.  In  fine,  the  contracted  curvature  tensor 
is  proportional  to  the  metrical  tensor  gu  .  For  a  three-space 
the  constant  factor  is—  2/R2,  and  for  a  four-manifold—  3/R*, 
and  so  on.  Notice  that  we  are  dealing  here  with  isotropic 
manifolds,  —  a  remark  which  will  be  of  importance  in  the 
sequel. 

The  curvature  invariant  is  G  =  guc  GM  ,  and  since  g™  gu  =  n, 
we  have,  by  (111), 


This  justifies,  in  general,  the  name  of  'mean  curvature' 
mentioned  in  Chapter  IV  and  given  to  G  by  some  authors. 
For  a  three-space  we  have 

G=-—   ,  (112s) 

R*  ' 

and  for  a  four-fold,  provided  always  it  were  isotropic,  we 
should  have 

G=-—  . 
R* 

It  was  known  for  a  long  time  that  the  line-element  of  a 
three-space  of  constant  curvature  l/R2  is,  in  polar  coordinates 

xi,  xt,  x3  =  r,  0,  6, 

d<r*  =  drz+R2  sin2—  .  [<fy2+sin2  0d0»].  (113) 

R 

In  fact,  availing  himself  of  (75),  the  reader  will  find  for  (113), 
as  the  only  surviving  components, 

Gii  =  -  —  ,   G22  =  -2  sin2  —  ,  G33  =  -  sin2  0  G«, 

/?2  R 

that  is  to  say> 

G«=--g,,,  (HI,) 


126  RELATIVITY  AND  GRAVITATION 

thus  verifying  (111)  for  the  case  w  =  3,  whence  also  G  = 
—  6/R2,  as  above. 

Manifestly,  if  we  took  for  d<r2  the  negative  of  (113),  or 

2 
inverted  the  signs  of  all  gik  ,   we  should  have  Git  =  +  —  ga- 

Now,  it  will  be  well  to  notice  that  the  same  is  the  case  if  we 
subtract  the  (113)  -value  of  d<r2  from  the  squared  differential 
of  a  fourth  coordinate  multiplied  by  a  constant;  that  is  to 
say,  for  a  four-dimensional  manifold  defined  by 

ds?  =  dx4*-dr*-R*sin*  —  .  [d0*+sinty#a]  (114) 

R 

we  have  [not  (111)  with  n  =  4  but] 


as  the  reader  can  verify  explicitly,  and  therefore, 


In  fine,  for  a  four-fold,  say  space-time,  of  the  type  (114)  the 
three  curvature  components  and  the  invariant  G  have  the 
same  values  as  for  an  isotropic  three-space  with  changed 
signs.  Notice  that  this  result  does  by  no  means  clash  with 
the  general  equations  (111)  and  (112).  For  the  space-time 
determined  by  the  line-element  (114)  is  not  isotropic  with 
respect  to  its  riemannian  curvature,  even  if  #4  be  replaced  by 

•N/^l  *4. 

The  latter  line-element  plays  an  important  rdle  in  Ein- 
stein's recently  modified  theory  of  which  a  brief  account  will 
be  given  in  Appendix,  B. 

Consider  the  four-fold  defined  by  the  somewhat  more 
general  line-element 

dsz  =  g4dx42-dr2-R2  sin2  — 

R 

where  g4,  written  for  g44,  is  a  function  of  r  alone.  Then,  with 
&4  =  log  g4,  the  only  surviving  G-components  will  be 


CONSTANT  CURVATURE 
2 


127 


—  G44=  -I  (fcY+  2ht")  -     i  cot  - 
£4  R  R 

whence    the   curvature  invariant   G=  —  Gn 
with  g2=-R2  sin2  (r/R), 


Let  us  now  require  that  G  should  be  constant  (which  is,  at  any 
rate  a  necessary  condition  for  Gtli  :  g^  —  const.)-  Then  the  last 
formula  will  be  a  differential  equation  for  /u  =  log  #4-  Now, 
this  equation  can  be  satisfied  by 

g4  =  cos2  ar, 
where  a  is  a  constant.    In  fact,  this  assumption  gives 

h't  =-2a  tan  ar,  h"4  =  -  2a2/cos2  ar 
and  reduces  the  last  equation  to 

4<z          r  6 

2a2+--  cot  -  .  tan  ar  =  G  —  —=  const., 
R          R  R2 

and  this  equation  can  only  be  satisfied  either  by  a  =  0,  i.e., 
g4  =  l,  and 

6 


which  leads  to  the  line-element  just  considered,  or  by  a  —  l/R, 
i.e.,  gt  =  cos2(r/R),  and  G  =  12/^2,  which  gives  the  line- 
element 

ds2  =  cos2-^-  •  ^42-^2-^2sin2  —  [</ 
R  R 


(116) 


utilized  by  de  Sitter.     (Cf.  Appendix,  C,  infra).    The  con- 


128  RELATIVITY  AND  GRAVITATION 

stant  value  of  the  invariant  G  is  in  this  case 

r    12 
G=R*' 

that  is  to  say,  apart  from  the  changed  sign,  such  as  would 
correspond,  by  (112),  to  a  genuine  isotropic  /owr-fold  con- 
sidered at  the  beginning.  Moreover,  introducing  g4  =  cos2(r/R) 
into  (115),  we  have  at  once 

/-»  O  ^,  ^-          x~»  <t      '     *>     ^         s~*  *-* 

"^'G22=^7G33=       >VG«==  ]p*. 

and  since  g\  —  —  1,  gz  =  —  ^2  sin2(r/.R),  and  all  components  with 
IT^K  vanish, 

C..  =  -|a.,  (116') 

which,  apart  from  the  changed  sign  of  the  constant  factor, 
agrees  with  (111)  for  n  =  4. 

On  the  other  hand,  substituting  into  (115)  the  alternative 
solution  g4  =  l  we  have,  for  the  line-element  (114), 

£«=  J-2  &*  (•=1.2.3);   ^44  =  0. 

The  best  way  of  stating  the  properties  of  the  two  solutions 
is  to  write  the  corresponding  contravariant  tensors  which  in 
our  case  reduce  to  Gtt=Gu/gu.  These  are,  for  the  line- 
element  (114), 

Gii  =  G»2  =  G3.8=  ^  f  G44  =  0, 
and  for  the  line-element  (116), 


Thus  the  time-space  defined  by  the  line-element  (116) 
behaves,  apart  from  the  common  sign  change,  as  an  orderly 
four-fold  of  constant  and  isotropic  riemannian  curvature. 
This  is  its  characteristic  difference  from  the  manifold  defined 
by  (114)  which  is  deprived  of  isotropy  and  is  a  rather  loose, 
uneven  melange  of  time  and  space.  Such  at  least  would  be 


EINSTEIN'S  NEW  EQUATIONS  129 

the  comparison  of  Einstein's  line-element  (114)  with  de  Sitter's, 
(116),  from  the  standpoint  of  general  geometry.  Their 
physical  merit  must,  of  course,  be  judged  by  other  standards. 

B.      Einstein's  New  Field-Equations  and  Elliptic  Space. 

About  two  years  after  the  publication  of  the  original  form 
of  the  gravitational  field-equations,  (III),  Chapter  IV, 
Einstein  found  weighty  reasons  for  slightly  modifying  them.* 
Without  attempting  an  exhaustive  discussion  of  all  his  reasons 
for  that  change  or  amplification  we  shall  give  here  a  brief 
account  of  his  new  field-equations  and  of  some  of  their 
consequences. 

The  tensor  of  matter  TM  being  given,  the  metrical  and 
at  the  same  time  the  gravitation  tensor  components  glK  are 
not,  of  course,  determined  by  the  field-equations  alone,  as 
indeed  would  be  the  case  with  any  other  set  of  differential 
equations  in  infinite  space  (and  time).  A  necessary  supple- 
ment of  the  data  consisted,  exactly  as  in  the  case  of  Laplace- 
Poisson's  equation,  in  prescribing  the  behaviour  of  the  gw  at 
infinity.  Now,  as  may  best  be  seen  from  the  example  of  the 
radially  symmetrical  field  treated  in  Chapter  V,  the  gllc  were 
assumed  to  tend  'at  infinity',  that  is,  for  ever  growing  r/L, 
to  their  galilean  values  gtK  ,  say  in  cartesian  coordinates, 

-1000 
0-1  0  0 
0  0-1  0 
0001 

But  such  boundary  or  limit  conditions,  not  being  independent 
of  the  choice  of  the  coordinate  system,  have  seemed  'repugnant 
to  the  spirit  of  the  relativity  principle'.  In  fact,  to  remain 
generally  invariant  the  limit  tensor  would  have  to  be  an  array 
of  sixteen  zeros.  Moreover,  the  adoption  of  the  galilean  or 
inertial  tensor  at  infinity  would  be  tantamount  to  giving 
up  the  requirement  of  the  relativity  of  inertia.  For  whereas 
the  inertia  or  mass  of  a  particle  generally  depends  upon  the 

*A.  Einstein,  Kosmologische  Betrachtungen  zur  allgemeinen  Relativitdts- 
theorie.  Berlin  Sitzungsberichte,  1917,  pp.  142-152. 


130  RELATIVITY  AND  GRAVITATION 

gw  and  these  are  even  at  the  surface  of  the  sun  but  slightly 
different  from  glK  ,  the  mass  of  the  particle  at  infinity  would 
differ  but  very  little  from  what  it  is  near  the  sun  or  other 
celestial  giants.  In  fine,  the  bulk  of  its  mass  would  be  inde- 
pendent of  other  bodies,  and  if  the  particle  existed  alone  in 
the  whole  universe,  it  would  still  retain  practically  all  its 
mass.  As  a  matter  of  fact  we  do  not  know  whether  such 
would  not  be  the  case.*  But  somehow,  not  uninfluenced  by 
Mach's  older  ideas,  Einstein  inclines  to  the  belief  that  every 
particle  owes  its  whole  inertia  to  all  the  remaining  matter  in 
the  universe.  Yet  another  reason  against  the  said  conditions 
at  infinity  is  given  which  is  based  on  considerations  borrowed 
from  the  statistical  theory  of  gases  and  which  would  equally 
apply  to  Newton's  theory.  But  for  this  the  reader  must  be 
referred  to  Einstein's  original  paper  (/.  c.,  §1). 

In  conclusion  Einstein  confesses  his  inability  to  build  up 
any  satisfactory  conditions  at  infinity,  in  space  that  is.f  But 
here  a  way  out  naturally  suggested  itself.  The  conditions  at 
infinity  being  hard  or  perhaps  impossible  to  find,  let  the 
world  or  universe  be  closed  in  all  its  space  extensions.  If  this 
be  a  possible  assumption,  no  such  conditions  were  needed. 

Thus  Einstein  comes  to  assume  space  to  be  a  finite,  closed 
three-fold  of  constant  curvature,  in  short  an  elliptic  space, 
either  of  the  antipodal  (spherical)  or  of  the  polar,  properly 
'elliptic',  kind.  But,  as  we  saw  before,  the  curvature  proper- 
ties of  space-time  are  modified  by  the  presence  of  matter,  the 
invariant  G,  for  instance,  being  proportional  to  the  density 
of  matter.  Thus  the  curvature  of  space,  as  a  section  of  the 
four-fold,  can  only  be  approximately  constant  and  isotropic, 
and  Einstein  assumes  therefore  that  space  is  elliptic  or  very 
nearly  so  on  the  whole,  deviating  here  and  there,  within  and 
near  condensed  matter,  from  the  average  value  of  its  curvature 
"L/R2  and  from  isotropy,  somewhat  as,  in  two  dimensions,  a 

*Provided,  of  course,  we  had  some  massless  phantoms  to  serve  us  as 
a  reference  system  and  thus  to  enable  us  to  state  the  lonely  particle's  perse- 
verance in  uniform  motion. 

t'Fiir  das  raumlich  Unendliche'.  There  is  nowhere  a  mention  of  the 
behaviour  at  infinite  past  or  future,  no  doubt,  because  such  questions  with 
regard  to  time  are  not  urgent  in  the  usual  (stationary)  type  of  problems. 


EINSTEIN'S  NEW  EQUATIONS  131 

slightly  corrugated  or  wrinkled  sphere.    As  we  know,  the  line- 
element  of  such  a  three-space  is 


R 

and  Einstein  constructs  the  line-element  which  is  to  determine 
the  four-  world  '  on  the  whole  '  by  simply  subtracting  dv2  from 


In  short,  far  enough  from  condensed  matter,  stars,  planets, 
and  so  on,  his  line-element,  in  polar  coordinates,  is 


-R2  sin2  —  (d<j>2+  sintyft*),  (114) 

R 

a  differential  form  treated  in  Appendix  A.* 

Now,  this  line-element  is  incompatible  with  Einstein's 
older  field-equations  (III).  In  fact,  the  corresponding  curva- 
ture tensor  consists  of  the  only  surviving  components 

G,,=     -g,-,,    G14  =  0;   G=,  (114') 


*From  the  four-dimensional  point  of  view,  the  assumption  that  three- 
dimensional  space  is  elliptic  is,  of  course,  as  unsatisfactory  as  the  older 
assumption  of  galilean  gtlc  at  infinity.  For  although  the  space  properties  as 
defined  by  dffz  are  invariant  for  transformations  of  the  Xi,  x^,  Xz  alone  into 
any  x'i,  x'z,  x'3,  they  cease  to  be  so  when  all  four  coordinates  are  freely 
transformed.  What  is  then  invariant  are  the  curvature  properties  of  the 
four-fold  of  which  the  three-space  is  an  arbitrary  section.  If  at  least  the 
four-fold  (114)  were  isotropic,  Einstein's  elliptic  space  could  be  invariantly 
defined  as  that  of  its  sections  to  which  corresponds  the  minimum  mean 
curvature,  and  this  is  the  mean  curvature  of  the  four-fold  itself  (cf.  W. 
Killing,  loc.  cit.,  pp.  79-83).  But  the  four-fold  defined  by  (114)  is  by  no 
means  isotropic,  as  was  explained  in  A.  Figuratively,  and  with  some 
licence,  it  resembles  not  a  sphere  but  rather  the  surface  of  a  circular 
cylinder.  By  (114)  not  only  the  value  of  the  curvature  of  three-space 
remains  unsettled  but  even  its  property  of  being  at  all  a  closed  space.  In 
fine,  the  assumption  that  three-space  is  elliptic  should  be  as  'repugnant  to 
the  spirit  of  relativity'  as  was  the  older  condition  at  infinity.  But  as  a 
matter  of  fact  it  did  not  appear  to  Einstein  in  that  light. 

The  clearest  way  of  stating  Einstein's  new  assumption  is  to  say  that, 
outside  of  condensed  matter,  it  is  possible  to  choose  a  coordinate  system  in 
which  the  line-element  ds*  assumes  the  form  (114). 


132  RELATIVITY  AND  GRAVITATION 

and  if  these  values  be  introduced  into  the  field-equations 
(Ilia),  which  are  identical  with  (III),  the  result  is 


But  'on  the  whole',  that  is,  outside  of  condensed  matter,  Tn, 
TM,  T33  are  to  vanish  (though  the  value  of  T44  and  T  =  p  need 
not  be  prejudiced),  and  since  actually  gn=  —  1,  etc.,  the  in- 
compatibility of  (114)  with  (III)  is  manifest. 

Such  being  the  case,  Einstein  is  driven  to  modify  his 
original  equations  (III)  by  subtracting  from  their  left-hand 
members  the  terms  Xgw  with  a  constant  X.  Thus  his  new 
field-equations  are 

Gu  -Xg«  =  -  -  (Tx  -  i  &.  T),  (117) 

£2 

and  since  these  give,  obviously, 

G-4\=  —  T,  (117a) 

c2 

they  can  also  be  written 

=~Tm.  (1176) 


Since  the  supplementary  term  \gtK  is  itself  covariant  of  rank 
two,  the  general  covariance  of  the  new  equations  is  manifest. 
It  remains  to  evaluate  the  constant  X  in  terms  of  the 
curvature  1/R2-.  Now,  if  we  assumed  that  outside  of  'con- 
densed matter  '  there  is  no  matter  at  all,  i.e.,  TIK  =  0  for  all  L,  K, 
we  should  have,  by  (1141)  and  the  first  of  (1176),  X  =  l/#2, 
clashing  with  (117a),  through  (1141)  which  would  require 
\  =  3/R2.  But,  as  Einstein  expressly  states,  his  new  theory 
is  to  be  associated  with  the  approximate  concept  that  all  the 
matter  of  the  universe  is  spread  uniformly  over  immense  spaces. 
In  other  words,  Einstein  substitutes  for  the  granular  structure 
of  the  universe  (the  grains  being  not  only  planets  but  stars, 
nebulae  and  similar  giants)  a  macroscopically  homogeneous 
distribution  of  matter,  exactly  as  for  many  purposes  a  con- 


EINSTEIN'S  NEW  EQUATIONS  133 

tinuous  homogeneous  medium  is  substituted  for  an  assem- 
blage of  molecules  or  atoms.  The  total  mass  contained  in  the 
universe  being  M  and  its  volume  V,  Einstein's  homogeneous 
density,  prevailing  on  the  whole,  is 

M 


Only  here  and  there,  within  the  celestial  bodies,  the  density  p 
exceeds  p0  considerably,  and  is  perhaps  somewhat  larger  in 
interstellar  spaces  within  our  galaxy  than  half  way  between 
one  star  cluster  or  'island  universe'  and  another,  a  million  or 
more  light  years  apart.  Moreover,  basing  himself  upon  the 
known  fact  of  the  small  relative  velocity  of  stars  as  compared 
with  the  light  velocity,  Einstein  makes  the  approximate 
assumption  that  there  is  a  coordinate  system,  relatively  to 
which  matter  is  on  the  whole  permanently  at  rest,  and  in 
which  therefore  the  tensor  of  matter  is  reduced  to  its  44-th 
component  which  is  then  also  its  invariant  T  =  p. 
In  fine,  we  have  outside  of  condensed  matter 


as  the  only  surviving  component,  and  therefore,  by  (1141) 
and  (1176), 

\  1  47T 

A  =   —    =  -  p0. 

R*       c* 

Thus,  Einstein's  new  field-equations    (117)  become  ulti- 
mately 

.=-87;r«.  (U7c) 

At  the  same  time  we  see  that  the  curvature  of  space  on  the 
whole  is  proportional  to  the  average  density  of  matter, 


The  whole  volume  of  elliptic  space  of  the  polar  or  properly 
elliptic  kind  being 


134  RELATIVITY  AND  GRAVITATION 

the  total  mass  of  the  universe,  in  astronomical  units,  will  be 

M=—R,  (119) 

4 

which  moved  some  authors  to  the  enthusiastic  exclamation: 
'the  more  matter,  the  more  room'.  The  corresponding 
'gravitation  radius',  or  better,  the  mass  in  bary-optical  units, 
which  is  a  length,  would  be 

L=^  =  ^,  (lift,) 

c2        4 

or  just  one-quarter  of  the  total  length  of  an  elliptic  straight 
line.* 

According  even  to  our  coarse  knowledge  of  the  average 
density  of  matter  (some  thousand  suns  per  cubic  parsec) ,  and 
in  view  of  the  formula  (118),  it  is  impossible  to  believe  in  a 
curvature  radius  much  smaller  than  1012  astronomical  units 
or,  say,  R  —  IQ20  kilometers.  This  would  mean,  by  (119a),  a 
total  mass  amounting  again,  in  bary-optical  units,  to  almost 
1020  kilometers.  To  this  tremendous  total  our  own  sun  contri- 
butes but  1|  kilometers,  and  our  whole  galaxy  not  more  than 
1010  kilometers.  The  total  would  thus  require  1010  such  galaxies 
or  Shapley 's  '  island  universes ' .  All  these  stellar  systems  may 
perhaps  be  found  among  the  spirals.  But  if  Shapley's  esti- 
mate (Bull.  Nat.  Res.  Council,  1921,  No.  11,  The  Scale  of  the 
Universe)  be  materially  correct,  these  island  universes  are 
from  500  thousand  to  10  million  light  years  apart,  and  then 
it  remains  to  be  seen  whether  the  last  mentioned  space  would 
be  ample  enough.  Yet  it  would  certainly  be  foolish  to  deny 
the  possibility  of  a  much  larger  R  and  of  the  existence  of 
many  more  island  universes.  That  Einstein's  requirement, 
at  least  in  the  present  state  of  astronomical  knowledge,  can 
at  any  rate  be  satisfied,  is  perhaps  best  seen  from  its  form 
(118)  which  is  compatible  with  as  small  an  average  density 
as  we  please. 

*The  total  length  of  a  straight  line  (geodesic)  in  the  polar  kind  of  space 
is  irR,  and  in  the  antipodal  or  spherical  kind  of  space  2wR.  The  total 
volume  of  the  latter  space  is  2TT2R3,  which  would  give  the  double  mass,  as 
in  Einstein's  paper.  The  space  in  question  being  thus  far  defined  only 
differentially,  the  choice  between  the  polar  and  antipodal  kind  remains  free. 


.  SITTER'S  SPACE-TIME  135 

Further  details  concerning  these  cosmological  speculations 
will  be  found  in  de  Sitter's  third  paper  on  Einstein's  Theory 
of  Gravitation,*  where  the  role  played  by  elliptic  space  in 
astronomy  since  the  time  of  Schwarzschild  (1900)  is  discussed. 

The  light  rays  corresponding  to  Einstein's  line-element 
(114)  turn  out  to  be  straight  lines  in  elliptic  space,  and  these 
lines,  described  with  uniform  velocity,  are  also  the  orbits  of 
free  particles.  Planetary  motion  would  undergo  some  modi- 
fications due  to  the  finite  value  of  R]  but  these  are,  for  the 
present,  too  small  to  be  detected.  Nor  does  Einstein's 
'cosmological  term',  as  the  supplement  gJR2  to  his  original 
field-equations  is  called,  lead  to  any  other  predictions  verifi- 
able in  our  days  by  experiment  or  observation. 

C.      Space-Time  according  to  de  Sitter. 

Returning  to  Einstein's  amplified  field-equations  (117) 
let  us  assume,  with  de  Sitter,  that  there  is  outside  of  'con- 
densed matter'  no  matter  at  all,  so  that  in  such  domains  all 
the  components  of  T^  ,  including  7*44,  vanish.  Thus  we  shall 
have,  in  free  space,  so  to  speak, 

G*  =  Xgu 

for  all  i,  K.  Now,  as  we  saw  in  Appendix  B,  these  equations > 
which  are  of  the  form  of  (111),  can  be  satisfied  by  the  line- 
element  (116),  and  give  G  =  12/R2.  And  since,  on  the  other 
hand, 

G  =  g"GlK  =  Xr^  =  4X, 

we  have  3 

~  R2' 

This  is  the  solution  of  the  cosmological  problem  proposed 
by  de  Sitter  in  his  last  quoted  paper.  Thus,  de  Sitter's  free 
space-time  is  defined  by  the  line-element 

<fc2  =  cos2  -   c2dt2  -  dr2  -  R2  sin2—  [d<t>2+sm2<j>dd2]          (116) 
R  R 

and  is  therefore,  as  we  saw,  a  manifold  of  constant  isotropic 
*W.  de  Sitter,  Monthly  Notices  of  R.A.S,,  November  1917. 


136  RELATIVITY  AND  GRAVITATION 

curvature.  Within  matter  Einstein's  new  equations,  with 
X  =  3/7?2,  are  valid,  i.e., 

Gu-%=-"*(^-  J&.r).  (120) 

J\.  C" 

The  isotropy  of  de  Sitter's  space-time,  expressed  by 
Gu  =  G22  =  G™  =  G**  =  —  , 

£2> 

as  in  (1162;),  distinguishes  it  characteristically  from  Einstein's 
space-time.  This  goes  hand  in  hand  with  p0  =  0  outside  of 
matter  proper. 

De  Sitter's  line-element  differs  from  Einstein's  by 

g44  =  COS2  — 

R 

instead  of  £44  =  !.  Consequently,  if  the  permanency  of  atoms 
be  assumed  as  in  Chapter  V,  the  spectrum  lines  of  distant 
stars  should  be  displaced  towards  the  red.  If  r  be  the  distance 
of  a  star  from  an  observer  placed  at  the  origin  of  coordinates, 
the  observed  wave-length  should  be  increased  from  1  to 

1  :  cos  —  ,  becoming  infinite  for  r  =  —  R,  the  greatest  distance 
R  2 

possible  in  a  properly  elliptic  space.  Manifestly,  everything 
is  at  a  standstill  at  the  equatorial  belt,  i.e.,  all  along  the  polar 
of  any  observing  station  as  pole.  This,  though  sounding 
strangely,  entails  no  actual  difficulty  at  all.  As  to  the  spec- 
trum shift  of  less  distant  celestial  objects,  de  Sitter  quotes 
the  helium  or  .B-stars  which  show  a  systematic  displacement 
towards  the  red  such  as  would  correspond  to  a  velocity  of 
4*5  km.  per  sec.  If,  as  de  Sitter  suggests,  one-third  of  this 
is  considered  as  a  gravitational  Einstein-effect,  the  remainder 
may  be  accounted  for  by  the  decrease  of  £44,  and  since  the 
average  distance  of  the  .B-stars  is  believed  to  be  r  =  3.107 
astronomical  units,  we  should  have 


R 

and   therefore   a   curvature  radius  ^  =  |1010.     But  there  is, 
for  the  present,  nothing  cogent  in  the  attribution  of  the  said 


GRAVITATION  AND  ELECTRONS  137 

remainder  of  spectrum  shift  to  the  dwindling  of  g44  with  mere 
distance,  and  it  would  certainly  be  premature  either  to  reject 
or  to  accept  the  results  of  this  attractive  piece  of  reasoning. 

Other  consequences  of  the  theory  and  a  more  thorough 
comparison  with  Einstein's  solution  will  be  found  in  de 
Sitter's  paper.  Here  it  will  be  enough  to  mention  still  that 
according  to  de  Sitter's  line-element  the  parallax  of  a  star 
should  reach  a  minimum  at  r  =  %irR,  the  greatest  distance  in 
the  polar  kind  of  space  (which  de  Sitter  prefers  to  the  anti- 
podal). This  minimum,  of  the  semi-parallax,  is  equal  to 
p=a/R,  if  a  be  the  distance  of  the  earth  from  the  sun.  On 
the  other  hand,  Einstein's  line-element  gives,  for  r  =  %irR,  a 
vanishing  parallax.  Since  de  Sitter's  minimum  is  at  least  as 
small  as  p  =  10"  10  =  0". 00002,  one  cannot  reasonably  hope  to 
discriminate  between  the  two  proposals  by  direct  observations 
of  parallaxes,  while  indirect  ones  contain  too  many  assump- 
tions to  be  considered  as  crucial. 

Soon  after  the  publication  of  de  Sitter's  paper  Einstein 
raised  some  objections  to  his  form  of  the  line-element.  For 
these,  however,  not  altogether  crushing,  the  reader  must  be 
referred  to  Einstein's  own  paper  (Berlin  Sitzungsberichte , 
March  1918,  pp.  270-272). 

D.     Gravitational  Fields  and  Electrons. 

The  problem  of  the  equilibrium  of  electricity  constituting 
the  electron  as  the  structural  element  of  matter  proper, 
already  attacked  by  G.  Mie  and  others,  has  been  taken  up 
by  Einstein  in  a  paper  of  April  1919  (Berlin  Sitzungsber., 
pp.  349-356).  The  result  of  the  investigation  is  that  this 
tempting  question  cannot  be  completely  answered  by  means 
of  the  field-equations  alone.  For  details  of  the  reasoning 
the  reader  must  be  referrred  to  the  original  paper.  It  will 
be  enough  to  mention  that  the  fixed  relation  between  the 
universal  constant  X  in  the  amplified  field-equations  and  the 
total  mass  of  the  universe,  as  related  in  Appendix  B,  is  here 
given  up.  Space  continues  to  be  considered  as  closed  but  the 
curvature  radius  R  and,  therefore,  the  volume  of  the  universe 
appears  as  independent  of  the  total  mass  contained  in  it, 
though  its  macroscopic  density  p0  is  still  treated  as  uniform. 


INDEX 


(The  numbers  refer  to  the  pages) 


Abraham,  M 75 

Absolute,  Cayley's 51,  52 

Absolute  differential  calculus. .  39 
Absolute  value,  or  size,  of  vec- 
tor   51 

Angle.. . ...  56 

Antipodal  kind  of  elliptic  space  6 

Antisymmetrical  tensors 44 

Associated  invariant 53 

vectors 54 

Astronomical  unit  of  mass ....  74 

Atoms,  as  natural  clocks 104 

1  Bary-optical  unit  of  mass 134 

Bessel 10 

Bianchi,  L 28,39,64,71 

Boundary  conditions 129 

Canal  rays 105 

Cayley 51,52 

Centrifugal  acceleration 31,37 

Christoffel 59 

symbols 27 

Clifford 13 

Closed  space 130 

Coelostat  distortions 102 

Compatibility  conditions 118 

Componens  of  a  tensor 41 

Conjugate  vectors 54 

Conservation  laws 87 

Constant  curvature 67 

Constant   light-velocity,    prin- 
ciple    2 

Contraction,  of  mixed  tensors.  46 

Contravariant  devriative 59 

tensor,  defined.  41 

Cosmological  term ; .  135 

Covariance  of  natural  laws 23 

Covariant  differentiation .....  59 

tensor,  defined 41 

Current,  four- 109 

Curvature,  gaussian 17,  65 

riemannian .  .  67 


Curvature  invariant 79, 125 

Curvature  tensor . 63, 68 

contracted 70 

Cyanogen  spectrum  lines 104 

Deflection  of  light  rays 100-102 

Density  of  matter 134 

Differential  geometry 5 

quadratic  form ...  14 
Differentiation  of  tensors .  .  59  et  seq. 
Divergence  of  mixed  tensor.  . .  83 

of  a  vector 62 

of  a  six- vector ....     62 

Eclipse  expeditions 102 

Eddington,  A.  S 39, 80, 103 

Einstein,  1, 10, 11,  12,  19,  22,  24,  28, 

38,  39,  56,  58,  61,  70,  77,  82,  86, 

88,  104,  107,  113,  129,  130,  137 

Electro-magnetic  six- vector .  .  .   109 

Electromagnetic  equations ....   106 

et  seq. 
stress,  momentum, 

and  energy 75 

Electrons 137 

Electrostatic  potential 112 

Elementary  flatness 13 

Elevator 11 

Elliptic  space 6, 130 

Energy,  principle  of 86-87 

Energy  tensor 75, 122 

EotvOs,  R 10 

Equation  of  motion,  of  a  free 

particle 28 

Equivalence  hypothesis 12 

Equivalent  differential  forms .  .     16 

Expansion,  of  a  vector 59 

of  a  six- vector ....     62 

Fermat's  principle 100-101 

Field   equations,   gravitational 

70,77,89,132 


UNIVERSITY  OF  CALIFORNIA 
DEPARTMENT  OF  CIVIL  ENGINEEI 
rERKELEY.  CALIFORNIA 


140 


INDEX 


Fixed-starssystem 1 

Four-current 109 

Four-index  symbols 17,  64 

Four-potential Ill 

Four- vector 4, 40 

Fundamental  quadratic  form .  .     52 

tensor 52 

Free  particle  motion,  and  geo- 
desies   8,  20 

Galaxies 134 

Galilean  coefficients 6, 129 

Galileo 10 

Gaussian  coordinates 39 

General  relativity  principle ...     22 

Geodesies 7,  26-28 

Gradient,  of  a  scalar  field 49 

Gravitation,     and     Christoffel 

symbols 29 

Gravitation  law,  Newton's. ...     73 

Gravitation  radius 95 

Gravitational   field   equations, 

69  et  seq.,  89, 132 
waves 90 

Hamiltonian  principle 88 

Heavy  and  inert  mass 10 

Helium  or  B-stars 136 

Hilbert,  D 88 

Holonomous  transformations. .  16 

Homaloidal,  or  flat,  manifold .  .  67 

Hydrogen  nucleus 80 

Indices,  upper  and  lower 45 

Inertia,  induced 130 

of  energy 74 

Inertial  systems 1 

Inner  product,  of  tensors 42 

Invariance,  of  line-element 16 

Invariants 42, 46, 47 

metrical 53, 57, 

62, 79, 88, 125 

Island  universes 134 

Isotropic  curvature 67, 125 


Jacobian.  .  . 
Jeffreys,  H.. 
Jewell,  E.L. 


15 
99 

104 


Keplerian  laws 96, 98 

Killing,  W 64,124,131 

Kottler,  F 107 

Laplace-Poisson's  equation .  .69,  73, 

74,79 

Laue,  M.  v 75, 76 

Law  of  (algebraic)  inertia 16 

Levi-Civita,  T 39, 41,101 

Light  propagation,  and  mini- 
mal lines 8, 20 

Line-element 5 

Linear  differential  form 113 

Lipschitz's  theorem 66 

Local  coordinates 12 

Lor,  matrix 75 

Lorentz,  H.  A 88 

Lorentz  transformation 4 

Mach,  E 130 

Magneto-electric  six- vector . . .    107 
Mass,  astronomical  unit  of. ...     74 

Matter 74,75,77 

equations 82, 85, 123 

Maxwell's  electromagnetic 

stress 122 

equations 106 

Mean  curvature 79, 125 

Mercury's  perihelion  motion .  .     99 

Metrical  manifold 50 

properties  of  tensors .      53 

Mie,  G 10,137 

Minimal  lines 7 

Minkowski 4,75 

Mixed  tensor,  defined 45 

Momentum,  principle  of 86-87 

Mosengeil,  K.  v 74 

Natural  clocks 104 

volume 58 

Newcomb 99 

Newton 10 

Newton's  equations  of  motion.  l?6 

Node  motion,  of  Venus 99 

Non-holonomous    transforma- 
tions   14 

Norm,  of  a  vector 53 


INDEX 


141 


Outer  product,  of  tensors 43 

Orthogonal  coordinates 113 

vectors,  defined.  .  .     56 

Parallax 137 

Perihelion  motion 95  et  seq. 

Permanent  field 70 

Perturbations,  secular 99 

Planetary  motion 95  et  seq. 

Polar  kind  of  elliptic  space  ...       6 
Ponderomotive  force,  in  elec- 
tromagnetic field 119 

Potential,  electrostatic 112 

four- Ill 

newtonian 36 

retarded 90 

vector- 112 

Poynting 74,  75 

Principe,  eclipse  expedition .  .  .   102 
Principles  of  momentum  and 

energy 86-87 

Product,  inner 42, 48 

outer 43 

Propagation  of  gravitation ....     90 
Proper  time 103 

Radially  symmetrical  field.  92  et  seq. 

Radius,  gravitation- 95, 134 

of  world-curvature .  . .  80-81 

Rank  ,of  a  tensor 40  et  seq. 

Rankine,  A.O.,  and  Silberstein,  117 

Reduced  tensor 56 

Retarded  potential 90 

Reference  frameworks .  .ltet  passim 
Relativity  principle,  general. .  .     22 

special 2 

Ricci,  G 39, 41 

Riemann 17,51,64 

Riemannian  manifold 50 

Riemann-Christoffel  tensor.  .62,  63 

Rotating  system 30-35, 101 

Rotation,  of  a  covariant  vector    61 

Russell,  H.  N 102 

Rutherford 81 

Scalar,  tensor  of  rank  zero ....     42 

Scalar  product,  of  tensors 42 

Schwarzschild,  K 95, 135 


Seeliger 99 

Shapley 134 

Shift  of  spectrum  lines.  102-105, 136 

Sitter,  W.  de.  .33,  99, 101, 127, 135, 

136, 137 

Six- vector 44 

Size,  of  a  vector 51 

Sobral,  eclipse  expedition 102 

Space-like  vector 53 

Special  relativity,  recalled ....    1-9 
Spectrum  shift,  due  to  gravita- 
tion  102-105 

due  to  distance 136 

Spherical  space. 6 

St.  John,  C.  E 104 

Stress-energy  tensor 75 

Sum  of  tensors 41 

Sun,  gravitation  radius  of. .  .95, 134 

Supplement  of  a  tensor 55 

Sylvester 16 

Symmetrical  tensors 43 

Tangential  world 13 

Tensors SQetseq. 

Tensor  character,  criterion  of. .     48 

Tensor  of  matter 76, 78 

Thirring,  H 32 

Time-like  vector 53 

Universe,  mass  and  volume  of .   134 

Vector 40 

Vector  potential 112 

Velocity  of  light 25, 118 

Venus,  motion  of  nodes 99 

Volume 58 

Wave,  electromagnetic,  in  gra- 
vitation field 118 

Waves,  gravitational 90 

Wave  surface .  26 

Water,  curvature  in 80 

Weight  and  mass 10 

Weyl,  H 39, 88, 103, 113 

World  curvature 80 

vector 4 

Wright,  J.  E 39 

Zodiacal  matter .  .  99 


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